Subtraction Of Matrix - MathsTips.com

In this article, we will introduce the operation on a matrix that is the subtraction of matrices. Subtraction between two matrixes is possible if they have the same order or the dimensions. So, they must have the same number of rows and columns in order to subtract two or more matrices. Subtraction of matrix is possible by subtracting the element of another matrix if they have the same order.

Suppose Mohan has two shops at places A and B. Each shop sells clothes for boys and girls in three different price categories. The quantities sell by each shop are represented as matrices given below:

Shop 1: $Boys\begin {bmatrix} 90\\65 \\85 \end {bmatrix}$ $Girls\begin{bmatrix} 50\\55 \\75 \end{bmatrix}$

Shop 2: $Boys\begin {bmatrix} 80\\75 \\90 \end {bmatrix}$ $Girls\begin{bmatrix} 90\\45 \\85 \end{bmatrix}$

Suppose Mohan wants to know the total of loss in each price categories in both the shops. So, this can be represented in the matrix form as $Boys-Girls=\begin {bmatrix}90-80 &50-90 \\ 65-75 &55-45 \\ 85-90 &75-85 \end {bmatrix}$ = $\begin {bmatrix} 10 &-40 \\ -10&10 \\5&-10 \end {bmatrix}$

This new matrix is the difference between the above two matrices. So, the difference between two matrices is obtained by subtracting the corresponding elements of the given matrices.

Let A and B be two matrices of the same order (m*n). Let $A=[a_{ij}]$ , $B=[b_{ij}]$ . Then A-B is a matrix of the same order as A and B and its element are obtained by subtracting the elements of B from the corresponding elements of A. Thus if C = $[c_{ij}]$ = A-B, then $[c_{ij}]$ = $[a_{ij}]$ – $[b_{ij}]$ .

If $A=\begin {bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\end {bmatrix}$ is a 2*2 matrix and $B=\begin {bmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\end {bmatrix}$ is another 2*2 matrix. Then, we define $A-B=\begin {bmatrix} a_{11}-b_{11} & a_{12}-b_{12}\\ a_{21}-b_{21} & a_{22}-b_{22}\end {bmatrix}$ .

Properties of subtraction of matrices

It is a non-commutative operation. If we reverse the order of the matrices and subtract both of them with the same order/dimensions, the result will differ. A-B $\neq$ B-A
The negative of matrix A is written as (-A) such that if the addition of matrix with the negative matrix will always produce a null matrix. A+(-A)=0

Conditions for subtraction of matrices

Two matrices should be of same order (number of rows=number of columns).
Add the corresponding element of other matrices.

Subtraction of matrix of order 2*2

Example 1:Let $A=\begin {bmatrix} 2 & 3\\ 4 & 1\end {bmatrix}$ and $B=\begin {bmatrix} 1 & 1\\ 3& 2\end {bmatrix}$ . Find A-B.

Solution : $A-B=\begin {bmatrix} 2-1 & 3-1\\ 4-3& 1-2\end {bmatrix}$
$A-B=\begin {bmatrix} 1 & 2\\ 1& -1\end {bmatrix}$

Subtraction of matrix of order 3*3

Example 2:Let $A=\begin {bmatrix} 5 & 4 &3 \\ 7&8 &9 \\ 3 &12 &-5 \end {bmatrix}$ and $B=\begin {bmatrix} 2&3&4 \\ 4&-5 &6 \\ 0&7 &8 \end {bmatrix}$ . Find A-B.

Solution: $A-B=\begin {bmatrix} 5-2&4-3&3-4 \\ 7-4&8-(-5) &9-6 \\ 3-0&12-7 &(-5)-8 \end {bmatrix}$

$A-B=\begin {bmatrix} 3&1&-1 \\ 3&13 &3 \\ 3&5 &-13 \end {bmatrix}$

Exercise

Let $A=\begin {bmatrix} 6 & 8\\ -14 & 33\end {bmatrix}$ and $B=\begin {bmatrix} 12 & 5\\ -27 & -8\end {bmatrix}$ Find A-B.
Let $A=\begin {bmatrix} 2 &3 \\ -1&4 \\ 1& 2 \end {bmatrix}$ and $B=\begin {bmatrix} 1 &4 \\ 2&1 \\ -1&-2 \end {bmatrix}$ . Find A-B.
Let $A=\begin {bmatrix} 2 & 4 & 3 \\ -3 & -1 & 0 \end {bmatrix}$ and $B=\begin {bmatrix} 1 & -2 & 3 \\ 2& 4 & 5 \end {bmatrix}$ Find A-B.
Let $A=\begin {bmatrix} 8 & 12 &5 \\ -3&19 &22 \\ -1 &5 &6 \end {bmatrix}$ , $B=\begin {bmatrix} 11 & 13 &1 \\ 10&-5 &11 \\ -8 &2 &7 \end {bmatrix}$ . Find A-B.
Let $A=\begin {bmatrix} 2 & 3 &-1 \\ -1&2 &0\end {bmatrix}$ , $B=\begin {bmatrix} 1 & -1 &4 \\ 2&1 &2\end {bmatrix}$ and $C=\begin {bmatrix} -1 & 2 &1 \\ 3&-1 &0\end {bmatrix}$ Verify that A-(B-C)=A-(B+C).
Let $A=\begin {bmatrix} 2 & 4\\ 6 &8\end {bmatrix}$ and $B=\begin {bmatrix} 6 & 9\\ -12 & 6\end {bmatrix}$ Verify that A-B $\neq$ B-A.

Properties of subtraction of matrices

Conditions for subtraction of matrices

Subtraction of matrix of order 2*2

Subtraction of matrix of order 3*3

Exercise

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