# Subtraction of Two Matrices

We will learn how to find the subtraction of two matrices.

If A and B two matrices of the same order then A – B is a matrix which is the addition of A and –B.

For Example:

Let A = $$\begin{bmatrix} 0 & 1\\ 4 & 5\\ 3 & 7 \end{bmatrix}$$ and B = $$\begin{bmatrix} 2 & -6\\ 8 & 4\\ 5 & -2 \end{bmatrix}$$

Then, A – B = A + (-B) = $$\begin{bmatrix} 0 & 1\\ 4 & 5\\ 3 & 7 \end{bmatrix}$$ + $$\begin{bmatrix} -2 & 6\\ -8 & -4\\ -5 & 2 \end{bmatrix}$$

= $$\begin{bmatrix} 0 - 2 & 1 + 6\\ 4 - 8 & 5 - 4\\ 3 - 5 & 7 + 2 \end{bmatrix}$$

= $$\begin{bmatrix} - 2 & 7\\ -4 & 1\\ -2 & 9 \end{bmatrix}$$

Note: The elements of A – B can also be obtained by subtracting the elements of B from the corresponding elements of A.

For Example:

Let A = $$\begin{bmatrix} 15 & -8\\ 6 & 1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 1 & 4\\ -1 & 3 \end{bmatrix}$$

Now subtracting the elements of B from the corresponding elements of A we get,

A – B = $$\begin{bmatrix} 15 & -8\\ 6 & 1 \end{bmatrix}$$ - $$\begin{bmatrix} 1 & 4\\ -1 & 3 \end{bmatrix}$$

= $$\begin{bmatrix} 15 - 1 & -8 - 4\\ 6 + 1 & 1 - 3 \end{bmatrix}$$

= $$\begin{bmatrix} 14 & -12\\ 7 & -2 \end{bmatrix}$$.

Solved Examples on Subtraction of Two Matrices:

1. If M = $$\begin{bmatrix} 2 & 5\\ -1 & 3 \end{bmatrix}$$ and B = $$\begin{bmatrix} 1 & 1\\ 4 & -2 \end{bmatrix}$$, find M – N.

Solution:

M – N = $$\begin{bmatrix} 2 & 5\\ -1 & 3 \end{bmatrix}$$ - $$\begin{bmatrix} 1 & 1\\ 4 & -2 \end{bmatrix}$$

= $$\begin{bmatrix} 2 & 5\\ -1 & 3 \end{bmatrix}$$ + $$\begin{bmatrix} -1 & -1\\ -4 & 2 \end{bmatrix}$$

= $$\begin{bmatrix} 2 - 1 & 5 - 1\\ -1 - 4 & 3 + 2\end{bmatrix}$$

= $$\begin{bmatrix} 1 & 4\\ -5 & 5\end{bmatrix}$$.

2. If X = $$\begin{bmatrix} 16 & -5\\ 4 & 1 \end{bmatrix}$$ and Z = $$\begin{bmatrix} -13 & 4\\ 2 & 0 \end{bmatrix}$$, find X – Z.

Solution:

X – Z = $$\begin{bmatrix} 16 & -5\\ 4 & 1 \end{bmatrix}$$ -  $$\begin{bmatrix} -13 & 4\\ 2 & 0 \end{bmatrix}$$

= $$\begin{bmatrix} 16 & -5\\ 4 & 1 \end{bmatrix}$$ + $$\begin{bmatrix} 13 & -4\\ -2 & 0\end{bmatrix}$$

= $$\begin{bmatrix} 16 + 13 & -5 - 4\\ 4 - 2 & 1 - 0\end{bmatrix}$$

= $$\begin{bmatrix} 29 & -9\\ 2 & 1\end{bmatrix}$$.