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.

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