Chapter 3 Matrices (Concepts)

Introduction to Matrices

A matrix is a rectangular arrangement of numbers or functions. These numbers or functions are called the elements or the entries of the matrix. Matrices are fundamental mathematical objects used to represent and manipulate data in a structured way.

Matrices are widely used in various fields of science, engineering, and mathematics, including solving systems of linear equations, computer graphics, quantum mechanics, economics, statistics, and machine learning. They provide a concise way to represent complex data and transformations.

Order of a Matrix

A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$ (read as 'm by n'). The number of rows ($m$) is always stated first, followed by the number of columns ($n$).

The element in the $i$-th row and $j$-th column of a matrix A is denoted by $a_{ij}$ or $A_{ij}$. The first subscript '$i$' indicates the row number, and the second subscript '$j$' indicates the column number.

A general $m \times n$ matrix A can be represented compactly as $A = [a_{ij}]_{m \times n}$. In expanded form, it is written as:

$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

Here, $a_{11}$ is the element in the first row and first column, $a_{12}$ is the element in the first row and second column, and so on, up to $a_{mn}$, which is the element in the $m$-th row and $n$-th column.

The total number of elements in an $m \times n$ matrix is the product of the number of rows and the number of columns, which is $m \times n$.

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Types of Matrices

Matrices are classified into various types based on their dimensions, the arrangement of their elements, and specific properties. Some common types are:

1. Column Matrix

A matrix that has only one column is called a column matrix. Its order is always of the form $m \times 1$, where $m$ is the number of rows.

Example: $A = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is a $3 \times 1$ column matrix. A column matrix is also sometimes called a column vector.

2. Row Matrix

A matrix that has only one row is called a row matrix. Its order is always of the form $1 \times n$, where $n$ is the number of columns.

Example: $B = \begin{bmatrix} 1 & -2 & 0 & 5 \end{bmatrix}$ is a $1 \times 4$ row matrix. A row matrix is also sometimes called a row vector.

3. Square Matrix

A matrix in which the number of rows is equal to the number of columns is called a square matrix.

An $m \times n$ matrix A is a square matrix if $m = n$. In this case, the matrix is said to be a square matrix of order 'n' or 'm'.

Example: $C = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ is a $2 \times 2$ square matrix of order 2.

The elements $a_{ij}$ where $i = j$ (i.e., $a_{11}, a_{22}, \dots, a_{nn}$) of a square matrix are called the diagonal elements or elements of the principal diagonal (also called the main diagonal). The diagonal elements lie along the line extending from the top left corner to the bottom right corner of the matrix.

4. Diagonal Matrix

A square matrix is called a diagonal matrix if all its non-diagonal elements are zero.

In a diagonal matrix $A = [a_{ij}]$, $a_{ij} = 0$ for all $i \neq j$. The diagonal elements $(a_{ii})$ can be zero or non-zero.

Example: $D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -3 \end{bmatrix}$ is a $3 \times 3$ diagonal matrix. The matrix $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ is also a diagonal matrix.

5. Scalar Matrix

A diagonal matrix is called a scalar matrix if all its diagonal elements are equal.

In a scalar matrix $A = [a_{ij}]$, $a_{ij} = 0$ for $i \neq j$ and $a_{ii} = k$ (where $k$ is a constant) for all $i$.

Example: $E = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$ is a $3 \times 3$ scalar matrix with $k=5$. A scalar matrix is a special case of a diagonal matrix.

6. Identity Matrix (Unit Matrix)

A square matrix in which all diagonal elements are 1 and all non-diagonal elements are 0 is called an identity matrix.

In an identity matrix $I = [a_{ij}]$, the elements are defined by:

$a_{ij} = \begin{cases} 1 & , & \text{if } i = j \\ 0 & , & \text{if } i \neq j \end{cases}$

An identity matrix is a special type of scalar matrix where the constant $k=1$.

An identity matrix of order $n$ is denoted by $I_n$ or simply $I$ when the order is clear from the context.

Example: $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ (order 2 identity matrix), $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ (order 3 identity matrix).

7. Zero Matrix (Null Matrix)

A matrix in which all elements are zero is called a zero matrix or null matrix.

A zero matrix of order $m \times n$ is denoted by $O$ or $O_{m \times n}$.

Example: $O_{2 \times 3} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ is a $2 \times 3$ zero matrix. The zero matrix is unique for a given order.

Equality of Matrices

Two matrices A and B are said to be equal if and only if they satisfy both of the following conditions:

1. They are of the same order. If matrix A is $m \times n$, then matrix B must also be $m \times n$.

2. Each element of A is equal to the corresponding element of B. This means that for all possible values of $i$ (row index) and $j$ (column index), the element $a_{ij}$ from matrix A must be equal to the element $b_{ij}$ from matrix B.

Mathematically, $A = B$ if and only if $A$ and $B$ have the same order, and $a_{ij} = b_{ij}$ for all $i$ and $j$.

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Operations on Matrices

Just like numbers, matrices can undergo various operations, such as addition, subtraction, and multiplication. However, these operations are defined under specific conditions related to the order (dimensions) of the matrices involved. Understanding these operations is crucial for using matrices to solve mathematical problems and model real-world scenarios.

Addition of Matrices

The operation of adding two matrices is defined only if both matrices have the same order. If the orders are different, their sum is not defined.

Let $A$ and $B$ be two matrices of the same order, say $m \times n$. $A = [a_{ij}]_{m \times n}$ and $B = [b_{ij}]_{m \times n}$.

Their sum, denoted by $A+B$, is a matrix $C = [c_{ij}]$ of the same order $m \times n$. The elements of the sum matrix $C$ are obtained by adding the corresponding elements of matrices A and B.

That is, for all $i$ (from 1 to $m$) and $j$ (from 1 to $n$), the element $c_{ij}$ in the $i$-th row and $j$-th column of $A+B$ is given by:

$c_{ij} = a_{ij} + b_{ij}$

In essence, matrix addition is performed element-wise.

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Properties of Matrix Addition

Matrix addition satisfies the following properties, assuming the matrices A, B, and C are all of the same order $m \times n$:

1. Commutative Law: Matrix addition is commutative. The order of addition does not affect the result.

$A+B = B+A$

This is because the addition of real or complex numbers (the elements) is commutative ($a_{ij} + b_{ij} = b_{ij} + a_{ij}$).

2. Associative Law: Matrix addition is associative. When adding three or more matrices, the grouping of matrices does not affect the result.

$(A+B)+C = A+(B+C)$

This follows from the associative property of addition of real or complex numbers.

3. Existence of Additive Identity: For any $m \times n$ matrix A, there exists a zero matrix $O_{m \times n}$ (a matrix of the same order with all elements being zero) such that:

$A+O = O+A = A$

The zero matrix $O_{m \times n}$ is the additive identity for matrices of order $m \times n$.

4. Existence of Additive Inverse: For every $m \times n$ matrix $A = [a_{ij}]$, there exists an additive inverse matrix, denoted by $-A = [-a_{ij}]$, such that:

$A+(-A) = (-A)+A = O$

The matrix $-A$ is obtained by multiplying each element of A by $-1$.

Scalar Multiplication

The operation of scalar multiplication involves multiplying a matrix by a single number (scalar).

If $A = [a_{ij}]_{m \times n}$ is a matrix and $k$ is a scalar (which can be a real or a complex number), then the scalar multiple $kA$ is a matrix obtained by multiplying every element of A by the scalar $k$.

$kA = [ka_{ij}]_{m \times n}$

The resulting matrix $kA$ has the same order as the original matrix A.

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Properties of Scalar Multiplication

Let A and B be matrices of the same order $m \times n$, and let $k, l$ be scalars. Scalar multiplication satisfies the following properties:

1. Distributivity over matrix addition:

$k(A+B) = kA + kB$

2. Distributivity over scalar addition:

$(k+l)A = kA + lA$

3. Associativity of scalar multiplication:

$(kl)A = k(lA)$

4. Multiplication by multiplicative identity scalar (1) and its inverse (-1):

$1 \cdot A = A$ and $(-1) \cdot A = -A$

5. Multiplication by zero scalar (0):

$0 \cdot A = O$ (Zero matrix of the same order as A).

Subtraction of Matrices

Similar to addition, the difference of two matrices $A$ and $B$ is defined only if they are of the same order.

The subtraction of matrix B from matrix A, denoted by $A-B$, can be defined as the addition of matrix A and the additive inverse of matrix B.

$A-B = A + (-1)B$

If $A = [a_{ij}]_{m \times n}$ and $B = [b_{ij}]_{m \times n}$, then $A-B$ is a matrix $D = [d_{ij}]_{m \times n}$ where:

$d_{ij} = a_{ij} - b_{ij}$

Just like addition, matrix subtraction is performed element-wise.

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Multiplication of Matrices

Matrix multiplication is a more complex operation than addition or subtraction, and it is not performed element-wise. The product of two matrices $A$ and $B$ is defined only under a specific condition on their orders.

Let $A = [a_{ij}]_{m \times n}$ be a matrix of order $m \times n$. Let $B = [b_{jk}]_{p \times q}$ be a matrix of order $p \times q$.

The product $AB$ is defined if and only if the number of columns in the first matrix (A) is equal to the number of rows in the second matrix (B).

Condition for $AB$ to exist: $n = p$

If this condition ($n=p$) is met, the product matrix, let's call it $C = AB$, will have the order $m \times q$.

Order of $A$ is $m \times n$. Order of $B$ is $n \times q$. Order of $AB$ is $m \times q$.

The element $c_{ik}$ in the $i$-th row and $k$-th column of the product matrix $C = AB$ is obtained by multiplying the elements of the $i$-th row of matrix A by the corresponding elements of the $k$-th column of matrix B and summing these products.

For the element $c_{ik}$ (in the $i$-th row and $k$-th column of $C$), we take the $i$-th row of A, which is $(a_{i1}, a_{i2}, \dots, a_{in})$, and the $k$-th column of B, which is $\begin{bmatrix} b_{1k} \\ b_{2k} \\ \vdots \\ b_{nk} \end{bmatrix}$. We then calculate the sum of the products of the corresponding elements:

This can be written using summation notation as:

The indices $j$ range from 1 to $n$, where $n$ is the number of columns in A (and rows in B).

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Properties of Matrix Multiplication

Assuming that the orders of the matrices are such that the indicated products and sums are defined, matrix multiplication satisfies the following properties:

1. Associative Law: Matrix multiplication is associative. For matrices A, B, and C where the products are defined:

$(AB)C = A(BC)$

2. Distributive Law: Matrix multiplication is distributive over matrix addition.

* Left Distributive Law: For matrices A, B, and C where the operations are defined:

$A(B+C) = AB + AC$

* Right Distributive Law: For matrices A, B, and C where the operations are defined:

$(A+B)C = AC + BC$

3. Existence of Multiplicative Identity: For every square matrix A of order $n \times n$, there exists an identity matrix $I_n$ of the same order such that:

$AI_n = I_n A = A$

The identity matrix $I_n$ acts as the multiplicative identity for square matrices of order $n$.

4. Multiplication by Zero Matrix: The product of any matrix with a zero matrix (of appropriate order for multiplication) is a zero matrix.

If A is an $m \times n$ matrix, then $A O_{n \times p} = O_{m \times p}$ and $O_{p \times m} A = O_{p \times n}$.

5. Non-Commutativity: In general, matrix multiplication is not commutative. That is, for two matrices A and B, $AB$ is generally not equal to $BA$. Even if both products $AB$ and $BA$ are defined and are of the same order (which happens if A and B are square matrices of the same order), $AB$ is usually not equal to $BA$.

6. Cancellation Law does not hold: Unlike multiplication of real numbers, if $AB = AC$ (where $A$ is not a zero matrix), it does not necessarily imply that $B=C$. Similarly, if $AB = O$ (where O is a zero matrix), it does not necessarily imply that either $A=O$ or $B=O$.

Transpose of a Matrix

The transpose of a matrix is a fundamental operation that rearranges the elements of a matrix by swapping its rows and columns. This operation has significant applications in various areas of mathematics and linear algebra.

Definition of Transpose

Let A be a matrix of order $m \times n$. The transpose of A, denoted by $A'$ or $A^T$, is the matrix obtained by interchanging the rows and columns of A.

If the original matrix is $A = [a_{ij}]_{m \times n}$, where $a_{ij}$ is the element in the $i$-th row and $j$-th column, then its transpose $A' = [a'_{ji}]_{n \times m}$ is an $n \times m$ matrix where the element $a'_{ji}$ in the $j$-th row and $i$-th column of $A'$ is equal to the element $a_{ij}$ in the $i$-th row and $j$-th column of A.

$a'_{ji} = a_{ij}$ for all $i$ and $j$.

In simpler terms, the first row of A becomes the first column of $A'$, the second row of A becomes the second column of $A'$, and so on.

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Properties of Transpose

The transpose operation satisfies several important properties. Let A and B be matrices and $k$ be a scalar (real or complex number). Assume the orders of the matrices are such that the indicated operations are defined.

1. Transpose of a Transpose: Taking the transpose of the transpose of a matrix gives back the original matrix.

$(A')' = A$

If $A = [a_{ij}]$, then $A' = [a_{ji}]$. The element in the $i$-th row, $j$-th column of $(A')'$ is the element in the $j$-th row, $i$-th column of $A'$, which is $a_{ij}$. So $(A')'_{ij} = a_{ij} = A_{ij}$.

2. Transpose of a Sum: The transpose of the sum of two matrices is the sum of their transposes. This property holds if A and B are of the same order.

$(A+B)' = A' + B'$

Let $A = [a_{ij}]$ and $B = [b_{ij}]$. Then $A+B = [a_{ij}+b_{ij}]$. $(A+B)' = [(a_{ij}+b_{ij})'] = [a_{ji}+b_{ji}]$. $A' = [a_{ji}]$ and $B' = [b_{ji}]$. $A'+B' = [a_{ji}+b_{ji}]$. Thus, $(A+B)' = A' + B'$.

3. Transpose of a Scalar Multiple: The transpose of a matrix multiplied by a scalar is equal to the scalar multiplied by the transpose of the matrix.

$(kA)' = kA'$

Let $A = [a_{ij}]$. Then $kA = [ka_{ij}]$. $(kA)' = [(ka_{ij})'] = [ka_{ji}]$. $A' = [a_{ji}]$. $kA' = [ka_{ji}]$. Thus, $(kA)' = kA'$.

4. Transpose of a Product: The transpose of the product of two matrices is the product of their transposes in reverse order. This is often called the "reversal law" for transpose of a product.

$(AB)' = B'A'$

This property holds if the product AB is defined.

Derivation of $(AB)' = B'A'$:

Let $A = [a_{ij}]_{m \times n}$ and $B = [b_{jk}]_{n \times p}$.

The product $AB = C = [c_{ik}]$ is an $m \times p$ matrix, where the element $c_{ik}$ in the $i$-th row and $k$-th column is given by the sum of products:

$\displaystyle c_{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$

The transpose $(AB)' = C'$ is a $p \times m$ matrix. The element in the $k$-th row and $i$-th column of $(AB)'$, denoted as $(AB)'_{ki}$, is equal to the element $c_{ik}$ from the original product matrix $C$.

Now consider the product $B'A'$. $B'$ is the transpose of $B$, so its order is $p \times n$, and its elements are $b'_{kj} = b_{jk}$. $A'$ is the transpose of $A$, so its order is $n \times m$, and its elements are $a'_{ji} = a_{ij}$.

The product $B'A'$ is defined because the number of columns in $B'$ ($n$) equals the number of rows in $A'$ ($n$). The order of the resulting matrix $B'A'$ is $p \times m$.

The element in the $k$-th row and $i$-th column of $B'A'$, denoted as $(B'A')_{ki}$, is obtained by multiplying the elements of the $k$-th row of $B'$ by the corresponding elements of the $i$-th column of $A'$ and summing the products.

$\displaystyle (B'A')_{ki} = \sum\limits_{j=1}^{n} (B')_{kj} (A')_{ji}$

Substitute the definitions of $b'_{kj}$ and $a'_{ji}$:

Since the multiplication of scalar elements is commutative ($b_{jk} a_{ij} = a_{ij} b_{jk}$), we can rewrite the sum in (ii):

Comparing the expression for $(AB)'_{ki}$ from (i) and $(B'A')_{ki}$ from the last step, we see that:

$(AB)'_{ki} = (B'A')_{ki}$ for all $k$ and $i$.

Since the matrices $(AB)'$ and $B'A'$ have the same order ($p \times m$) and their corresponding elements are equal, we conclude that:

$(AB)' = B'A'$

Symmetric and Skew-Symmetric Matrices

These are special classes of square matrices defined based on their relationship with their transpose.

Symmetric Matrix

A square matrix $A$ is called a symmetric matrix if its transpose is equal to the matrix itself.

$A' = A$

This condition implies that for all indices $i$ and $j$, the element in the $i$-th row and $j$-th column is equal to the element in the $j$-th row and $i$-th column ($a_{ij} = a_{ji}$). The matrix is symmetric about its main diagonal.

Example: $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$. Here, $a_{12}=2$ and $a_{21}=2$, so $a_{12}=a_{21}$. $a_{13}=3$ and $a_{31}=3$, so $a_{13}=a_{31}$. $a_{23}=5$ and $a_{32}=5$, so $a_{23}=a_{32}$. Also, $a_{11}=1, a_{22}=4, a_{33}=6$. If we take the transpose of A, we get $A' = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$, which is equal to A.

Skew-Symmetric Matrix

A square matrix $A$ is called a skew-symmetric matrix if its transpose is equal to the negative of the matrix.

$A' = -A$

This condition implies that for all indices $i$ and $j$, the element in the $i$-th row and $j$-th column is the negative of the element in the $j$-th row and $i$-th column ($a_{ij} = -a_{ji}$).

Consider the diagonal elements, where $i=j$. For a skew-symmetric matrix, $a_{ii} = -a_{ii}$. This equation can be rewritten as $2a_{ii} = 0$, which implies $a_{ii} = 0$. Therefore, all the diagonal elements of a skew-symmetric matrix must be zero.

Example: $B = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}$. Here, the diagonal elements are $b_{11}=0, b_{22}=0, b_{33}=0$. Also, $b_{12}=2$ and $b_{21}=-2$, so $b_{12} = -b_{21}$. $b_{13}=-3$ and $b_{31}=3$, so $b_{13} = -b_{31}$. $b_{23}=4$ and $b_{32}=-4$, so $b_{23} = -b_{32}$. If we take the transpose of B, we get $B' = \begin{bmatrix} 0 & -2 & 3 \\ 2 & 0 & -4 \\ -3 & 4 & 0 \end{bmatrix}$, which is equal to $-B$.

Theorem on Symmetric and Skew-Symmetric Matrices

Theorem: Any square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Proof:

Let A be any square matrix. We want to show that A can be written as the sum of a symmetric matrix and a skew-symmetric matrix.

Consider the expression:

Let $P = \frac{1}{2}(A+A')$ and $Q = \frac{1}{2}(A-A')$.

We will first show that P is a symmetric matrix. To do this, we need to find the transpose of P and show it is equal to P ($P'=P$).

$P' = \left(\frac{1}{2}(A+A')\right)'$

Using the property $(kA)' = kA'$, we get:

$P' = \frac{1}{2}(A+A')'$

Using the property $(A+B)' = A'+B'$, we get:

$P' = \frac{1}{2}(A' + (A')')$

Using the property $(A')' = A$, we get:

$P' = \frac{1}{2}(A' + A)$

Since matrix addition is commutative ($A'+A = A+A'$):

Since $P' = P$, P is a symmetric matrix.

Next, we will show that Q is a skew-symmetric matrix. To do this, we need to find the transpose of Q and show it is equal to -Q ($Q'=-Q$).

$Q' = \left(\frac{1}{2}(A-A')\right)'$

Using the property $(kA)' = kA'$ and $(A-B)' = A'-B'$ (which can be derived from $(A+B')'$ and $(kA)'$), we get:

$Q' = \frac{1}{2}(A-A')'$

$Q' = \frac{1}{2}(A' - (A')')$

Using the property $(A')' = A$, we get:

$Q' = \frac{1}{2}(A' - A)$

Factor out -1:

$Q' = -\frac{1}{2}(A - A')$

Since $Q = \frac{1}{2}(A-A')$, we have:

Since $Q' = -Q$, Q is a skew-symmetric matrix.

From (i), we expressed A as the sum of P and Q. From (ii), we showed P is symmetric. From (iii), we showed Q is skew-symmetric. Therefore, any square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

Uniqueness: To show the uniqueness, assume that A can also be expressed as the sum of a symmetric matrix S and a skew-symmetric matrix K, i.e., $A = S+K$, where $S'=S$ and $K'=-K$.

Taking the transpose of $A = S+K$:

$A' = (S+K)' = S' + K'$

Using the properties of S and K:

$A' = S + (-K) = S - K$

So we have two equations:

$A = S + K$

$A' = S - K$

Adding these two equations:

$A + A' = (S+K) + (S-K) = 2S$

This gives $S = \frac{1}{2}(A+A')$, which is the symmetric part P we found earlier.

Subtracting the second equation from the first:

$A - A' = (S+K) - (S-K) = 2K$

This gives $K = \frac{1}{2}(A-A')$, which is the skew-symmetric part Q we found earlier.

Since S must be $\frac{1}{2}(A+A')$ and K must be $\frac{1}{2}(A-A')$, the representation is unique.

Answer:

Elementary Operations on a Matrix

Elementary operations, also known as elementary transformations, are a set of basic operations that can be applied to the rows or columns of a matrix. These operations are fundamental tools used to simplify matrices, solve systems of linear equations, find the inverse of a matrix, and calculate the rank of a matrix. They are called "elementary" because they represent basic manipulations that do not change the essential properties of the linear system represented by the matrix.

There are three types of elementary row operations and three corresponding elementary column operations, totaling six elementary operations.

Elementary Row Operations

The three elementary row operations that can be performed on a matrix are:

1. Interchange of any two rows: This operation swaps the positions of two rows in the matrix.

Notation: The interchange of the $i$-th row and the $j$-th row is denoted by $R_i \leftrightarrow R_j$.

Example: Apply the operation $R_1 \leftrightarrow R_2$ to the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_2} \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix}$

2. Multiplication of the elements of any row by a non-zero scalar: This operation multiplies every element in a specific row by the same non-zero number. Multiplying by zero would make the row a zero row, which can alter the rank.

Notation: The multiplication of the elements of the $i$-th row by a non-zero scalar $k$ is denoted by $R_i \to k R_i$.

Example: Apply the operation $R_1 \to 2 R_1$ to the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \xrightarrow{R_1 \to 2 R_1} \begin{bmatrix} 2 \times 1 & 2 \times 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 3 & 4 \end{bmatrix}$

3. Addition of the corresponding elements of any other row multiplied by a non-zero scalar: This operation adds a scalar multiple of one row to another row. The original row being added to is replaced by the result of this addition.

Notation: The addition of $k$ times the elements of the $j$-th row to the corresponding elements of the $i$-th row is denoted by $R_i \to R_i + k R_j$. Note that the row $R_j$ itself remains unchanged.

Example: Apply the operation $R_2 \to R_2 + (-3) R_1$ (or $R_2 \to R_2 - 3 R_1$) to the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \xrightarrow{R_2 \to R_2 - 3 R_1} \begin{bmatrix} 1 \\ 3 + (-3)(1) & 4 + (-3)(2) \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3-3 & 4-6 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 0 & -2 \end{bmatrix}$

Elementary Column Operations

Similar to row operations, there are three elementary column operations performed on the columns of a matrix:

1. Interchange of any two columns: This operation swaps the positions of two columns in the matrix.

Notation: The interchange of the $i$-th column and the $j$-th column is denoted by $C_i \leftrightarrow C_j$.

Example: Apply the operation $C_1 \leftrightarrow C_2$ to the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \xrightarrow{C_1 \leftrightarrow C_2} \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}$

2. Multiplication of the elements of any column by a non-zero scalar: This operation multiplies every element in a specific column by the same non-zero number $k$.

Notation: The multiplication of the elements of the $i$-th column by a non-zero scalar $k$ is denoted by $C_i \to k C_i$.

Example: Apply the operation $C_2 \to 5 C_2$ to the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \xrightarrow{C_2 \to 5 C_2} \begin{bmatrix} 1 & 5 \times 2 \\ 3 & 5 \times 4 \end{bmatrix} = \begin{bmatrix} 1 & 10 \\ 3 & 20 \end{bmatrix}$

3. Addition of the corresponding elements of any other column multiplied by a non-zero scalar: This operation adds a scalar multiple of one column to another column. The original column being added to is replaced by the result of this addition.

Notation: The addition of $k$ times the elements of the $j$-th column to the corresponding elements of the $i$-th column is denoted by $C_i \to C_i + k C_j$. Note that the column $C_j$ itself remains unchanged.

Example: Apply the operation $C_2 \to C_2 + 2 C_1$ to the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \xrightarrow{C_2 \to C_2 + 2 C_1} \begin{bmatrix} 1 & 2 + 2(1) \\ 3 & 4 + 2(3) \end{bmatrix} = \begin{bmatrix} 1 & 2+2 \\ 3 & 4+6 \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 3 & 10 \end{bmatrix}$

It is important to note that when solving a problem (like finding the inverse), you should generally stick to either only row operations or only column operations throughout the process, but not mix them.

Equivalent Matrices

When a matrix B is obtained from a matrix A by performing one or a sequence of elementary operations (row operations or column operations), matrix B is said to be equivalent to matrix A. This is denoted by $A \sim B$.

Elementary operations preserve certain properties of the matrix, such as its rank. Equivalent matrices have the same rank. They are also crucial for solving systems of linear equations, as performing elementary row operations on the augmented matrix of a system of equations results in an equivalent system with the same set of solutions. Similarly, elementary column operations are related to operations on the variables of the corresponding linear system.

Invertible Matrices

The concept of the inverse of a matrix is analogous to the concept of a reciprocal in the set of real numbers. For a non-zero real number $a$, its reciprocal is $\frac{1}{a}$ such that $a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = 1$ (the multiplicative identity). Similarly, for a square matrix A, we seek a matrix, called its inverse, which when multiplied by A yields the identity matrix (the multiplicative identity for matrices).

Definition of Invertible Matrix and Inverse

A square matrix $A$ of order $n \times n$ is said to be invertible (or non-singular) if there exists a square matrix $B$ of the same order $n \times n$ such that the matrix products $AB$ and $BA$ are both equal to the identity matrix of order $n$, denoted by $I_n$.

$AB = BA = I_n$

In this case, the matrix B is called the inverse of A and is uniquely denoted by $A^{-1}$. So, if A is invertible, $AA^{-1} = A^{-1}A = I_n$.

A square matrix that is not invertible is called a singular matrix.

A key condition for a square matrix A to be invertible is that its determinant must be non-zero. This condition, $|A| \neq 0$, is a powerful criterion for checking invertibility and is studied in detail in the chapter on Determinants. Conversely, if $|A|=0$, the matrix is singular and its inverse does not exist.

Uniqueness of the Inverse

A significant property of the inverse matrix is that if a square matrix has an inverse, that inverse is unique. There cannot be two different matrices that are inverses of the same matrix.

Proof of Uniqueness:

Let A be an invertible matrix of order $n$. Suppose, for the sake of contradiction, that A has two different inverses, say B and C.

By the definition of an inverse, if B is an inverse of A, then:

Similarly, if C is an inverse of A, then:

Consider the matrix B. We can write B by multiplying it with the identity matrix, since $I_n$ is the multiplicative identity for matrices:

Now, using the definition of inverse from (ii), we can substitute $I_n$ with $AC$:

Matrix multiplication is associative, so we can change the grouping of the product:

Now, using the definition of inverse from (i), we can substitute $BA$ with $I_n$:

Finally, using the property of the identity matrix ($I_nC = C$):

This result, $B=C$, contradicts our initial assumption that B and C were two different inverses. Therefore, the initial assumption must be false, and an invertible matrix can have only one unique inverse.

Inverse of a Product of Matrices

If A and B are two invertible matrices of the same order $n$, then their product $AB$ is also an invertible matrix, and the inverse of the product $AB$ is the product of the inverses of B and A in reverse order.

$(AB)^{-1} = B^{-1}A^{-1}$

This property is also known as the reversal law for the inverse of a product, similar to the transpose of a product property.

Proof:

Let A and B be invertible matrices of order $n$. By definition, their inverses $A^{-1}$ and $B^{-1}$ exist and satisfy:

$AA^{-1} = A^{-1}A = I_n$

$BB^{-1} = B^{-1}B = I_n$

To show that $B^{-1}A^{-1}$ is the inverse of $AB$, we need to demonstrate that the product of $AB$ and $B^{-1}A^{-1}$ in both orders results in the identity matrix $I_n$.

Consider the product $(AB)(B^{-1}A^{-1})$. Using the associative property of matrix multiplication, we can regroup the matrices:

We know that $BB^{-1} = I_n$:

Using the property that $AI_n = A$:

Finally, using the property $AA^{-1} = I_n$:

Now consider the product $(B^{-1}A^{-1})(AB)$. Using the associative property:

We know that $A^{-1}A = I_n$:

Using the property that $I_nB = B$:

Finally, using the property $B^{-1}B = I_n$:

From (i) and (ii), we have shown that $(AB)(B^{-1}A^{-1}) = I_n$ and $(B^{-1}A^{-1})(AB) = I_n$. By the definition of an inverse matrix, this proves that $B^{-1}A^{-1}$ is the inverse of $AB$.

Therefore, $(AB)^{-1} = B^{-1}A^{-1}$.

Finding Inverse using Elementary Operations

One powerful method to find the inverse of a square matrix, if it exists, is by using elementary row operations (or elementary column operations). This method is based on the idea that performing an elementary row operation on a matrix A is equivalent to pre-multiplying A by a corresponding elementary matrix. An elementary matrix is obtained by performing the same elementary row operation on an identity matrix of the same order.

If a square matrix A of order $n$ can be transformed into the identity matrix $I_n$ by a sequence of elementary row operations, say $E_1, E_2, \dots, E_k$, then A is invertible. The sequence of operations can be represented as:

$E_k (\dots (E_2 (E_1 A)) \dots) = I_n$

This can be written concisely as $(E_k \dots E_2 E_1) A = I_n$. By the definition of the inverse, the matrix product $(E_k \dots E_2 E_1)$ is the inverse of A.

$A^{-1} = E_k \dots E_2 E_1$

To find $A^{-1}$ using this method, we apply the same sequence of elementary row operations to the identity matrix $I_n$.

$E_k (\dots (E_2 (E_1 I_n)) \dots) = E_k \dots E_2 E_1 = A^{-1}$

We combine these two processes by writing the augmented matrix $[A | I_n]$ (matrix A augmented with the identity matrix $I_n$) and performing elementary row operations simultaneously on both sides of the vertical line. The goal is to transform the left side (A) into the identity matrix ($I_n$). As the left side is transformed to $I_n$, the right side ($I_n$) is simultaneously transformed into $A^{-1}$.

$[A | I_n] \xrightarrow{\text{Elementary Row Operations}} [I_n | A^{-1}]$

Alternatively, one can use elementary column operations. In this case, we write the augmented matrix in a vertical format $[ \frac{A}{I_n} ]$ and apply elementary column operations to transform the top part (A) into the identity matrix ($I_n$). The bottom part ($I_n$) will then be transformed into $A^{-1}$.

$[ \frac{A}{I_n} ] \xrightarrow{\text{Elementary Column Operations}} [ \frac{I_n}{A^{-1}} ]$

It is crucial to apply either only row operations throughout the process or only column operations throughout the process, but not mix them. If, at any stage during the application of elementary row (or column) operations, the matrix on the left side (A) reduces to a matrix with a row (or column) of all zeros, then the matrix A is singular, and its inverse does not exist.

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