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