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.


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