Find the value of x if
We have the matrix equation,
We need to find the value of x.
Let us compute L.H.S:
Let,
and
In order to multiply two matrices, A and B, the number of columns in A must equal the number of rows in B. Thus, if A is an m x n matrix and B is an r x s matrix, n = r.
First, let us compute
Multiply 1st row of matrix A by matching members of 1st column of matrix B, then sum them up.
(1, x, 1).(1, 2, 15) = (1 × 1) + (x × 2) + (1 × 15)
⇒ (1, x, 1).(1, 2, 15) = 1 + 2x + 15
⇒ (1, x, 1).(1, 2, 15) = 2x + 16
Multiply 1st row of matrix A by matching members of 2nd column of matrix B, then sum them up.
(1, x, 1).(3, 5, 3) = (1 × 3) + (x × 5) + (1 × 3)
⇒ (1, x, 1).(3, 5, 3) = 3 + 5x + 3
⇒ (1, x, 1).(3, 5, 3) = 5x + 6
Multiply 1st row of matrix A by matching members of 3rd column of matrix B, then sum them up.
(1, x, 1).(2, 1, 2) = (1 × 2) + (x × 1) + (1 × 2)
⇒ (1, x, 1).(2, 1, 2) = 2 + x + 2
⇒ (1, x, 1).(2, 1, 2) = x + 4
So,
Now, compute
Multiply 1st row of matrix D by matching members of 1st column of matrix C, then sum them up.
(2x + 16, 5x + 6, x + 4).(1, 2, x) = ((2x + 16) × 1) + ((5x + 6) × 2) + ((x + 4) × x)
⇒ (2x + 16, 5x + 6, x + 4).(1, 2, x) = (2x + 16) + (10x + 12) + (x2 + 4x)
⇒ (2x + 16, 5x + 6, x + 4).(1, 2, x) = x2 + 2x + 10x + 4x + 16 + 12
⇒ (2x + 16, 5x + 6, x + 4).(1, 2, x) = x2 + 16x + 28
So, we have got
Now, put L.H.S = R.H.S
[x2 + 16x + 28] = [0]
This means,
x2 + 16x + 28 = 0
⇒ x2 + 14x + 2x + 28 = 0
⇒ x(x + 14) + 2(x + 14) = 0
⇒ (x + 2)(x + 14) = 0
⇒ (x + 2) = 0 or (x + 14) = 0
⇒ x = -2 or x = -14
Thus, x = -2, -14.