# Subtraction of Matrices

Subtraction of Matrices: The difference of two matrices is possible only when the order of the two matrices is the same. Similarly, addition of two matrices is done only when the order of the two matrices is the same. Thus, the resulting matrix will be of the same order.

Note: Subtraction of matrices of different order is not defined.

The arithmetic operations on matrices are possible based on their order. Like the addition and subtraction of matrices requires the same order, but for multiplication of matrices, we need to check if the number of columns of one matrix is equal to the number of rows of the second matrix. The order of matrix multiplication is the order of the resulting matrix.

## Definition of Subtraction of Matrices

Mathematically, if there are two matrices, say A = [aij] and B = [bij] of the same order, say m × n, then the subtraction of A and B, i.e., A – B is defined as:

Matrix D = [dij]

A – B = aij – bij

Thus,

dij = aij – bij, (i = 1,2,3,… and j= 1,2,3…)

D = A – B = aij – bij

Thus, the two matrices whose difference is calculated have the same number of rows and columns. The subtraction of the two matrices can also be defined as addition of A and -B (negative of matrix B), since the process of addition is similar to subtraction.

A – B = A + (-B)

### Subtraction of 2 x 2 Matrices

Suppose A and B are 2 x 2 matrices, such that;

$A=\left[\begin{array}{lll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] \text { and } B=\left[\begin{array}{lll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right]$

Then, subtraction of matrices A and B, will be given as:

$A-B=\left[\begin{array}{lll} a_{11}-b_{11} & a_{12}-b_{12} \\ a_{21}-b_{21} & a_{22}-b_{22} \end{array}\right]$

Fact: If A and B are two matrices of the same order, then;

A – B ≠ B – A

Thus, commutative law is not applicable for subtraction of matrices.

### Subtraction of 3 x 3 Matrices

Suppose A and B are 3 x 3 matrices, such that;

$A=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right] \text { and } B=\left[\begin{array}{lll} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{array}\right]$

Then, subtraction of matrices A and B, will be given as:

$A-B=\left[\begin{array}{lll} a_{11}-b_{11} & a_{12}-b_{12} & a_{13}-b_{13} \\ a_{21}-b_{21} & a_{22}-b_{22} & a_{23}-b_{23} \\ a_{31}-b_{31} & a_{32}-b_{32} & a_{33}-b_{33} \end{array}\right]$

## Solved Examples

Q.1: If A and B are two matrices. Then find subtraction of matrices A and B.

$A = \left(\begin{array}{ll} 3 & 5 \\ 9 & 8 \end{array}\right)$

And

$B = \left(\begin{array}{ll} 1 & 3 \\ 8 & 9 \end{array}\right)$

Solution:

$A – B = \left(\begin{array}{ll} 3 & 5 \\ 9 & 8 \end{array}\right)-\left(\begin{array}{ll} 1 & 3 \\ 8 & 9 \end{array}\right)$ $A – B = \left(\begin{array}{ll} 3 -1 &5 -3\\ 9-8 & 8-9 \end{array}\right)$ $A – B = \left(\begin{array}{ll} 2 & 2\\ 1 & -1 \end{array}\right)$

Q.2: $\text { If } A=\left[\begin{array}{ccc} 3 & 1 & 2 \\ 2 & -8 & 1 \\ 1 & -2 & 1 \end{array}\right] and~ B=\left[\begin{array}{ccc} -1 & 0 & 1 \\ 3 & 4 & 1 \\ 5 & -1 & 0 \end{array}\right] then ~find ~A-B.$

Solution: Let A – B = C.

C = A – B

C = $\left[\begin{array}{ccc} 3 & 1 & 2 \\ 2 & -8 & 1 \\ 1 & -2 & 1 \end{array}\right]-\left[\begin{array}{ccc} -1 & 0 & 1 \\ 3 & 4 & 1 \\ 5 & -1 & 0 \end{array}\right]$

C = $\left[\begin{array}{ccc} 3-(-1) & 1-0 & 2-1 \\ 2-3 & -8-4 & 1-1 \\ 1-5 & -2-(-1) & 1-0 \end{array}\right]$

C = $\left[\begin{array}{ccc} 4 & 1 & 1 \\ -1 & -12 & 0 \\ -4 & -1 & 1 \end{array}\right]$

### Practice Questions

1. Find 3A – B, if $A = \left[\begin{array}{ccc} 3 & 2 & 1 \\ 6 & 5 & 4 \\ 9 & 8 & 7 \end{array}\right] and~ B =\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right]$

2. What is the subtraction of two identity matrices, of the order 2 x 2?

3. What is the subtraction of two zero matrices of the same order?

## Frequently Asked Questions on Subtraction of Matrices

### What is subtraction of matrices?

Two matrices are subtracted, only if the order of the two matrices are the same. If the order is different, then subtraction of the two matrices are not defined.

### How to subtract two matrices?

Subtraction of two matrices with the same order is done, by subtracting the elements of the same positions in the matrices. If matrix A = aij and matrix B = bij, where i and j can be any value, then, A – B = aij – bij.

### How to subtract 2×2 matrices?

Suppose, A and B are the two matrices of order 2 x 2, then the difference between the two matrices will result in difference between the elements of matrices of the same positions, such that;

• a11 – b11
• a12 – b12
• a22 – b22
• a21 – b21

### Can we subtract the 3×3 matrix from the 2×2 matrix?

We cannot subtract the 3 x 3 matrix from the 2 x 2 matrix,because both the matrices have different order or dimensions. For subtraction of two matrices, their order should be the same.

### If A and B are matrices and K is a scalar, then what is the value of K(A – B)?

Since K is a scalar value, therefore it can be distributed inside the bracket, such that;
K (A – B) = KA – KB