Matrix Operations

What is a Matrix?

A matrix is a rectangular arrangement of numbers, symbols, or expressions written in rows and columns. It is a fundamental concept in linear algebra used to represent data, solve equations, and perform various mathematical operations.

Below is an example of a matrix structure of 3 rows and 4 columns:

$\left[\begin{array}{llll}a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34}\end{array}\right]_{3 \times 4}$

In general form, the above matrix is represented by $A=\left[a_{i j}\right]$

$a_{11}$, $a_{12}$,.. etc. are called the elements of the matrix.

$a_{ij}$ belongs to the ith row and jth column and is called the $(i,j)$ th element of the matrix.

Operations on Matrices

The addition, subtraction, and multiplication of matrices are the three basic algebraic matrix operations.

Condition for performing Addition and Subtraction operations:

• The order of the matrix should be identical for performing addition and subtraction operations.

Condition for performing Multiplication operations

• The first matrix's number of rows and the second matrix's number of columns should be the same.

Addition of matrices

Matrix addition is a basic operation where two matrices of the same order are added by combining their corresponding elements. If $A = [a_{ij}]$ and $B = [b_{ij}]$ are two matrices of order $m \times n$, then their sum is

$A + B = [a_{ij} + b_{ij}]$

This means each element of the resulting matrix is obtained by adding the elements in the same position from $A$ and $B$.

Rules and Conditions for Matrix Addition

For two matrices to be added:

1. Both must have the same order (same number of rows and columns).

   • Example: A $2 \times 3$ matrix can only be added to another $2 \times 3$ matrix.

2. Each element is added position-wise.

   • $(i, j)$-th element of the result = $a_{ij} + b_{ij}$.

3. Matrix addition follows these properties:

   • Commutative law: $A + B = B + A$

   • Associative law: $(A + B) + C = A + (B + C)$

Step-by-Step Method with Example

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}$

Step 1: Check if both matrices are of the same order.

• $A$ and $B$ are both $2 \times 3$ matrices → valid for addition.

Step 2: Add corresponding elements:

$A + B = \begin{bmatrix} 1+6 & 2+5 & 3+4 \\ 4+3 & 5+2 & 6+1 \end{bmatrix}$

Step 3: Simplify each entry:

$A + B = \begin{bmatrix} 7 & 7 & 7 \\ 7 & 7 & 7 \end{bmatrix}$

Thus, the sum of matrices $A$ and $B$ is another $2 \times 3$ matrix where all entries are $7$.
Example:
$
\begin{aligned}
\mathrm{A} & =\left[\begin{array}{lll}
10 & 20 & 30 \\
20 & 30 & 40 \\
30 & 40 & 50
\end{array}\right], \quad \mathrm{B}=\left[\begin{array}{lll}
50 & 40 & 30 \\
40 & 30 & 20 \\
30 & 20 & 10
\end{array}\right] \\
\mathrm{A}+\mathrm{B} & =\left[\begin{array}{lll}
10+50 & 20+40 & 30+30 \\
20+40 & 30+30 & 40+20 \\
30+30 & 40+20 & 50+10
\end{array}\right]=\left[\begin{array}{lll}
60 & 60 & 60 \\
60 & 60 & 60 \\
60 & 60 & 60
\end{array}\right]
\end{aligned}
$

Properties of matrix addition:

i) Matrix addition is commutative, A + B = B + A

ii) Matrix addition is associative, A + (B+C) = (A+B) + C

iii) Additive identity exists, which means there exists a matrix O (null matrix) such that A + O = A = O + A (Here O has the same order as A)

iv) Existence of additive inverse means there exists a matrix B such that A + B = O = B + A

v) Cancellation property: If A + B = A + C then B = C

If A + C = B + C then A = B

Note: All matrices taken in the above property explanation have the same order which is m × n.

Subtraction of matrices

Matrix subtraction is the process of subtracting corresponding elements of two matrices of the same order. If $A = [a_{ij}]$ and $B = [b_{ij}]$ are two matrices of order $m \times n$, then their difference is

$
\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}, \mathrm{B}=\left[\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}} \text { Then, } \mathrm{A}-\mathrm{B}=\left[\mathrm{a}_{\mathrm{ij}}-\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n} \text { for all } \mathrm{i}, \mathrm{j}}
$

Rules and Conditions for Matrix Subtraction

1. Both matrices must have the same order.

   • Example: A $3 \times 2$ matrix can only be subtracted from another $3 \times 2$ matrix.

2. Subtraction is done element-wise:

   • $(i,j)$-th element of result = $a_{ij} - b_{ij}$.

3. Matrix subtraction is not commutative:

   • $A - B \neq B - A$ (in general).

4. Subtraction is related to addition:

   • $A - B = A + (-B)$, where $-B$ is the negative of matrix $B$.

Example:

$A = \begin{bmatrix} 5 & 7 \\ 2 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 6 & 2 \end{bmatrix}$

Then

$A - B = \begin{bmatrix} 5-1 & 7-3 \\ 2-6 & 4-2 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ -4 & 2 \end{bmatrix}$
$
\begin{aligned}
\mathrm{A} & =\left[\begin{array}{lll}
10 & 20 & 30 \\
20 & 30 & 40 \\
30 & 40 & 50
\end{array}\right], \quad \mathrm{B}=\left[\begin{array}{lll}
50 & 40 & 30 \\
40 & 30 & 20 \\
30 & 20 & 10
\end{array}\right] \\
\mathrm{A}-\mathrm{B} & =\left[\begin{array}{lll}
10-50 & 20-40 & 30-30 \\
20-40 & 30-30 & 40-20 \\
30-30 & 40-20 & 50-10
\end{array}\right]=\left[\begin{array}{ccc}
-40 & -20 & 0 \\
-20 & 0 & 20 \\
0 & 20 & 40
\end{array}\right]
\end{aligned}
$

Properties of matrix Subtraction:

i) Matrix subtraction is not commutative, $A-B \neq B-A$
ii) Matrix subtraction is not associative, $A-(B-C) \neq(A-B)-C$

iii) Cancellation property:

If A - B = A - C then B = C

If A - C = B - C then A = B

Note: All matrices taken in the above property explanation have the same order which is m × n.

Matrix Multiplication

Matrix multiplication is a fundamental operation where rows of the first matrix are multiplied with columns of the second matrix. If $A$ is of order $m \times n$ and $B$ is of order $n \times p$, then their product $C = AB$ is of order $m \times p$, where each element is defined as:

$c_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}$

Types of Matrix Multiplication

1. Row by Column Multiplication

   • Standard matrix multiplication.

   • Possible only when the number of columns in $A$ equals the number of rows in $B$.

2. Example: If $A$ is $2 \times 3$ and $B$ is $3 \times 2$, then $AB$ is $2 \times 2$.

3. Scalar Multiplication

   • Each element of a matrix is multiplied by a scalar (a constant number).

   • If $k$ is a scalar and $A = [a_{ij}]$, then $kA = [k \cdot a_{ij}]$.

Rules and Properties of Matrix Multiplication

1. Order rule: If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$.

2. Matrix multiplication is not commutative: $AB \neq BA$ (in general).

3. It is associative: $(AB)C = A(BC)$.

4. It is distributive: $A(B+C) = AB + AC$.

Step-by-Step Examples with Solutions

Example 1: Row by Column Multiplication

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$

Step 1: Multiply row of $A$ with column of $B$:

$AB = \begin{bmatrix} 1 \cdot 2 + 2 \cdot 1 & 1 \cdot 0 + 2 \cdot 3 \\ 3 \cdot 2 + 4 \cdot 1 & 3 \cdot 0 + 4 \cdot 3 \end{bmatrix}$

Step 2: Simplify:

$AB = \begin{bmatrix} 4 & 6 \\ 10 & 12 \end{bmatrix}$

Example 2: Scalar Multiplication

If $k = 3$ and $A = \begin{bmatrix} 1 & -2 \\ 0 & 5 \end{bmatrix}$,

then

$3A = \begin{bmatrix} 3 & -6 \\ 0 & 15 \end{bmatrix}$

Let us understand about the types of matrices in detail with supporting examples.

Scalar multiplication:

Let $\mathrm{k}$ be any scalar number, and $A=\left[a_{i j}\right]_{m \times n}$ be a matrix. Then the matrix is obtained by multiplying every element $\mathrm{A}$ by a scalar $\mathrm{k}$ and denoted as kA.
$
\begin{aligned}
& k A=\left[k a_{i j}\right]_{m \times n} \\
& \qquad \mathrm{~A}=\left[\begin{array}{ll}
2 & 6 \\
3 & 7 \\
5 & 8
\end{array}\right] \text { then, } 3 \mathrm{~A}=\left[\begin{array}{ll}
3 \times 2 & 3 \times 6 \\
3 \times 3 & 3 \times 7 \\
3 \times 5 & 3 \times 8
\end{array}\right]=\left[\begin{array}{cc}
6 & 18 \\
9 & 21 \\
15 & 24
\end{array}\right]
\end{aligned}
$

Properties of scalar multiplication:

If $A$ and $B$ are two matrices and $k, l$ are scalar then
i) $k(A+B)=k A+k B$
ii) $k l(A)=k(I A)=l(k A)$
iii) $(k+I) A=k A+I A$
iv) $(-k) A=-(k A)=k(-A)$
v) $1 \mathrm{~A}=\mathrm{A},(-1) \mathrm{A}=-\mathrm{A}$

Note: $A$ and $B$ have the same order $m \times n$.

Row by Column Multiplication

Product AB can be found if the number of columns in matrix A and the number of Product $A B$ can be found if the number of columns in matrix $A$ and the number of rows in matrix B are equal. Otherwise, multiplication AB is not possible.

i) $A B$ is defined only if $\operatorname{col}(A)=\operatorname{row}(B)$
ii) $B A$ is defined only if $\operatorname{col}(B)=\operatorname{row}(A)$

If $\begin{aligned} & \mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}} \\ & \mathrm{B}=\left[\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{p}} \\ & \mathrm{C}=\mathrm{AB}=\left[\mathrm{c}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{p}} \\ & \text { Where } c_{\mathrm{ij}}=\sum_{\mathrm{j}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{ij}} \mathrm{b}_{\mathrm{jk}}, 1 \leq \mathrm{i} \leq \mathrm{m}, 1 \leq \mathrm{k} \leq \mathrm{p} \\ & =a_{i 1} b_{1 k}+a_{i 2} b_{2 k}+a_{i 3} b_{3 k}+\ldots+a_{i n} b_{n k} \\ & \end{aligned}$

Properties of matrix multiplication:

i) Multiplication may or may not be commutative, so AB may or may not be equal to BA.
ii) Matrix multiplication is associative, meaning $A(B C)=(A B) C$
iii) Matrix multiplication is distributive over addition, mean $A(B+C)=A B+A C$ and $(B+C) A=B A+C A$
iv) If matrix multiplication of two matrices gives a null matrix then it doesn't mean that any of those two matrices was a null matrix.

Example:
$
A=\left[\begin{array}{ll}
0 & 2 \\
0 & 0
\end{array}\right] \text { and } B=\left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right] \text {, then } A B=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right]
$
v) Matrix multiplication $A \times A$ is represented by $A^2$. Thus, $A \cdot A \cdot A \cdot A \ldots \ldots . . n$ times $=A^n$.
vi) if $A$ is $m \times n$ matrix then, $I_m A=A=A I_n$.

Solved Examples Based On Matrix Operations

Example 1: Find A+B if
$
A=\left[\begin{array}{rrr}
-3 & 2 & 4 \\
8 & 3 & 4
\end{array}\right], B=\left[\begin{array}{lll}
4 & 1 & 5 \\
1 & 0 & 2
\end{array}\right]
$

Solution:
As the orders of the matrices A and B are the same $(2 \times 3)$, we can add them
$
\begin{aligned}
& A+B=\left[\begin{array}{rrr}
-3+4 & 2+1 & 4+5 \\
8+1 & 3+0 & 4+2
\end{array}\right] \\
& A+B=\left[\begin{array}{lll}
1 & 3 & 9 \\
9 & 3 & 6
\end{array}\right]
\end{aligned}
$

Hence, the value of
$
A+B \text { is }\left[\begin{array}{lll}
1 & 3 & 9 \\
9 & 3 & 6
\end{array}\right]
$

Example 2: Find A - B if
$
A=\left[\begin{array}{lll}
8 & 6 & 5 \\
5 & 6 & 1
\end{array}\right], B=\left[\begin{array}{lll}
5 & 3 & 4 \\
2 & 4 & 0
\end{array}\right]
$

Solution:
As the order of both matrices are same $(2 \times 3)$, we can subtract them
$
\begin{aligned}
& A=\left[\begin{array}{lll}
8 & 6 & 5 \\
5 & 6 & 1
\end{array}\right], B=\left[\begin{array}{lll}
5 & 3 & 4 \\
2 & 4 & 0
\end{array}\right] \\
& A-B=\left[\begin{array}{lll}
8-5 & 6-3 & 5-4 \\
5-2 & 6-4 & 1-0
\end{array}\right] \\
& A-B=\left[\begin{array}{lll}
3 & 3 & 1 \\
3 & 2 & 1
\end{array}\right]
\end{aligned}
$

Hence, the value of
$
A-B \text { is }\left[\begin{array}{lll}
3 & 3 & 1 \\
3 & 2 & 1
\end{array}\right]
$

Example 3: If $\mathrm{X}$ and $\mathrm{Y}$ are two matrices such that
$
X+2 Y=\left[\begin{array}{ll}
5 & 2 \\
8 & 9
\end{array}\right]_{\text {and }}
$
$X-Y=\left[\begin{array}{cc}2 & -1 \\ 2 & 0\end{array}\right]$, then find the matrix $\mathbf{Y}$.
Solution:
Subtract both the given matrices
$
\begin{aligned}
& (X+2 Y)-(X-Y)=\left[\begin{array}{ll}
5 & 2 \\
8 & 9
\end{array}\right]-\left[\begin{array}{cc}
2 & -1 \\
2 & 0
\end{array}\right] \\
& \Rightarrow 3 Y=\left[\begin{array}{ll}
3 & 3 \\
6 & 9
\end{array}\right] \\
& \Rightarrow Y=\left[\begin{array}{ll}
1 & 1 \\
2 & 3
\end{array}\right]
\end{aligned}
$

Hence, the matrix $\mathrm{Y}$ is $\left[\begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right]$

Example 4: Which of the following ordered matrices A allow the addition A+A'=B

1) $A_{2\times 3}$

2) $A_{3\times 2}$

3) $A_{3\times 3}$

4) $A_{1\times 2}$

Solution:

Addition of Matrices -

Only matrices of the same order can be added or subtracted

When A has order 3x3, A' has order 3x3 as well.

Hence, the answer is option 3.

List of topics related to Matrix Operations

Matrix operations are related with several important concepts in linear algebra that extend their use in problem solving. Below are the key topics related to matrix operations.

NCERT Resources

NCERT resources are very helpful for understanding matrix operations in Class 12 Mathematics. They provide clear explanations, solved examples, and practice questions for better exam preparation. Below are the key NCERT resources for Chapter 3 – Matrices.

NCERT Class 12 Maths Notes for Chapter 3 - Matrices

NCERT Solutions for Class 12 Maths for Chapter 3 - Matrices

NCERT Exemplar Class 12 Maths Solutions for Chapter 3 - Matrices

Practice Questions based on Matrix Operations

Practice questions are essential to strengthen your understanding of matrix operations like addition, subtraction, and multiplication. They help in applying concepts and improving problem-solving speed. Below are practice questions and MCQs based on matrix operations.

Matrix Operations - Practice Question MCQ

You can practice questions on the related topics using the links shared below: