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

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