If \(x\begin{bmatrix}x&2\\3&1\end{bmatrix} + 2\begin{bmatrix}3&x\\-x&2\end{bmatrix} = \begin{bmatrix}5x&12\\3&7\end{bmatrix}\), then the value of x is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3
Multiplying each matrix by its scalar and adding:
\( \displaystyle \begin{bmatrix}x^2&2x\\3x&x\end{bmatrix} + \begin{bmatrix}6&2x\\-2x&4\end{bmatrix} = \begin{bmatrix}x^2 + 6&4x\\x&x + 4\end{bmatrix} \)
By equality of matrices, each element equals the matching element on the right.
\(4x = 12\), so \(x = 3\)
This value fits the other elements too:
\(x^2 + 6 = 15 = 5x\)
\(x = 3\)
\(x + 4 = 7\)
Why the other options are incorrect
Option (1) 2 — Incorrect. It satisfies \(x^2 + 6 = 5x\) only; the element \(4x = 8\) is not 12.
Option (2) \(-3\) — Incorrect. Then \(4x = -12\), not 12.
Option (4) 6 — Incorrect. Then \(4x = 24\), not 12.