# Negative of a Matrix

We will discuss about Negative of a Matrix.

The negative of the matrix A is the matrix (-1)A, written as – A.

For example:

Let A = $$\begin{bmatrix} 12 & -17\\ -5 & 9 \end{bmatrix}$$.

Then –A = (-1) $$\begin{bmatrix} 12 & -17\\ -5 & 9 \end{bmatrix}$$ = $$\begin{bmatrix} -12 & 17\\ 5 & -9 \end{bmatrix}$$

Clearly, the negative matrix is obtained by changing the signs of each element.

Solved examples on Negative of a Matrix:

1. If A = $$\begin{bmatrix} 2 & 5\\ 1 & 3 \end{bmatrix}$$ then find the negative matrix of A.

Solution:

A = $$\begin{bmatrix} 2 & 5\\ 1 & 3 \end{bmatrix}$$

The negative matrix of A = -A

Now by changing the signs of each element of matrix A

We get $$\begin{bmatrix} -2 & -5\\ -1 & -3 \end{bmatrix}$$

Therefore, the negative matrix of A = -A = $$\begin{bmatrix} -2 & -5\\ -1 & -3 \end{bmatrix}$$.

2. If M = $$\begin{bmatrix} 5 & -1\\ -3 & 2 \end{bmatrix}$$ then find the negative matrix of M.

Solution:

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

The negative matrix of M = -M

Now by changing the signs of each element of matrix M

We get $$\begin{bmatrix} -5 & 1 \\ 3 & -2 \end{bmatrix}$$

Therefore, the negative matrix of A = -A = $$\begin{bmatrix} -5 & 1 \\ 3 & -2 \end{bmatrix}$$.

3. If I = $$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$ then find -I.

Solution:

I = $$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$

The negative matrix of I = -I

Now by changing the signs of each element of matrix M

We get $$\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$$

Therefore, the negative matrix of I = -I = $$\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$$.

Note: A + (-A) = 0; i.e., Sum a matrix and its negative matrix = 0.