Subtraction of Matrices with Examples

Here you will learn subtraction of matrices with examples.

Let’s begin –

Subtraction of Matrices

Let two matrices A and B of the same order, the subtraction of matrix B from matrix A is denoted by A – B and is defined as A – B = A + (-B).

Note : If matrices are not of same order then subtraction of matrix is not possible.

Subtraction of matrices of order \(2\times 2\)

If  A = \(\begin{bmatrix} 2 & 1 \\ 0 &  2  \end{bmatrix}\) and B = \(\begin{bmatrix} 1 & 0  \\ 1 &  2  \end{bmatrix}\), then find A – B.

A – B = A + (-B) = \(\begin{bmatrix} 2 & 1 \\ 0 &  2  \end{bmatrix}\) – \(\begin{bmatrix} 1 & 0  \\ 1 &  2  \end{bmatrix}\)

 = \(\begin{bmatrix} 1 & 1  \\ -1 &  0  \end{bmatrix}\)

Subtraction of matrices of order \(3\times 3\)

If  A = \(\begin{bmatrix} 2 & 3 & 4 \\ 0 &  4 & 6 \\ 5 & 8 & 9 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & 0 & 5  \\ 5 &  3 & 2 \\ 0 & 4 & 7 \end{bmatrix}\), then find 3A – 2B.

We have, 3A – 2B = 3A + (-2)B

\(\implies\) 3A – 2B = \(\begin{bmatrix} 6 & 9 & 12 \\ 0 &  12 & 18 \\ 15 & 24 & 27 \end{bmatrix}\) -\(\begin{bmatrix} -6 & 0 & -10 \\ -10 &  -6 & -4 \\ 0 & -8 & -14 \end{bmatrix}\)

 = \(\begin{bmatrix} 0 & 9 & 2 \\ -10 &  6 & 14 \\ 15 & 16 & 13 \end{bmatrix}\)

Example : If  A = \(\begin{bmatrix} 2 & 3 & -5 \\ 1 & 2 & -1 \end{bmatrix}\) and B = \(\begin{bmatrix} 0 & 5 & 1  \\ -2 & 7 & 3 \end{bmatrix}\), then find A + B and A – B.

Solution : Clearly A and B both are matrices of the same order \(2\times 3\). So, A + B and A – B both are defined.

Now, A + B = \(\begin{bmatrix} 2 & 3 & -5 \\ 1 & 2 & -1 \end{bmatrix}\) + \(\begin{bmatrix} 0 & 5 & 1  \\ -2 & 7 & 3 \end{bmatrix}\)

= \(\begin{bmatrix} 2 & 8 & -4 \\ -1 & 9 & 2 \end{bmatrix}\)

and A – B = A + (-B) = \(\begin{bmatrix} 2 & 3 & -5 \\ 1 & 2 & -1 \end{bmatrix}\) + (-1)\(\begin{bmatrix} 0 & 5 & 1  \\ -2 & 7 & 3 \end{bmatrix}\)

= \(\begin{bmatrix} 2 & -2 & -6 \\ 3 & -5 & -4 \end{bmatrix}\)

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