# Equality of Matrices Definition with Examples

Here you will learn equality of matrices definition with examples.

Let’s begin –

## Equality of Matrices

Definition : Two matrice A = $$[a_{ij}]_{m\times n}$$ and B = $$[b_{ij}]_{r\times s}$$ are equal if

(i) m = r i.e. the number of rows in A equals the number of rows in B.

(ii) n = s i.e the number of columns in A equals the number of columns in B.

(iii) $$a_{ij}$$ = $$b_{ij}$$ for i = 1, 2, ……. , m and j = 1, 2, ,,,,, , n.

If two matrices A and B are equal, we write A = B, otherwise we write A $$\ne$$ B.

The matrices A = $$\begin{bmatrix} 3 & 2 & 1 \\ x & y & 5 \\ 1 & -1 & 4 \end{bmatrix}$$ and B = $$\begin{bmatrix} 3 & 2 & 1 \\ -1 & 0 & 5 \\ -1 & -1 & z \end{bmatrix}$$ are equal if x = -1, y = 0 and z = 4.

Matrices $$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$ and $$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ are not equal, because their orders are not same.

Example : Find the value of x, y, z and w which satisfy the matrix equation, $$\begin{bmatrix} x – y & 2x + z \\ 2x – y & 3z+ w \end{bmatrix}$$ = $$\begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$$

Solution : Since the corresponding elements of two equal matrices are equal. Therefore,

$$\begin{bmatrix} x – y & 2x + z \\ 2x – y & 3z+ w \end{bmatrix}$$ = $$\begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$$

$$\implies$$ x – y = -1, 2x + z = 5, 2x – y = 0, 3z + w = 13

Solving the equation x – y = -1 and 2x- y = 0 as simultaneous linear equations, we get x = 1 and y = 2.

Now, putting x = 1 in 2x + z = 5, we get z = 3. Substituting z = 3 in 3z + w = 13, we obtain w = 4

Thus, x = 1, y = 2, z = 3 and w = 4