Let \( \mathrm{A}=\begin{bmatrix}3x & -2\\-10 & 2x\end{bmatrix} \) and \( \mathrm{B}=\begin{bmatrix}5 & 4\\2 & 3\end{bmatrix} \) Then which of the following are TRUE?
(A) \( |A|=|B| \) if \( x=\pm\displaystyle\frac{3}{\sqrt{2}} \)
(B) B is a symmetric matrix
(C) A is a singular matrix for \( x=\pm\sqrt{\displaystyle\frac{10}{3}} \)
(D) B is an invertible matrix
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
Every statement depends on the two determinants:
\( \displaystyle |\mathrm{A}|=(3x)(2x)-(-2)(-10)=6x^2-20 \)
\( \displaystyle |\mathrm{B}|=15-8=7 \)
For |A| = |B|:
\( \displaystyle 6x^2-20=7 \)
\( \displaystyle x^2=\frac{27}{6}=\frac{9}{2} \)
A is singular when |A| = 0:
\( \displaystyle 6x^2=20 \)
\( \displaystyle x^2=\frac{10}{3} \)
So |A| = |B| when \( x=\pm\frac{3}{\sqrt{2}} \), A is singular when \( x=\pm\sqrt{\frac{10}{3}} \), and |B| = 7 ≠ 0.
Checking the statements
(A) \( |A|=|B| \) if \( x=\pm\frac{3}{\sqrt{2}} \) — Correct. \( x^2=\frac{9}{2} \) gives \( x=\pm\frac{3}{\sqrt{2}} \).
(B) B is a symmetric matrix — Incorrect. The (1, 2) entry is 4 but the (2, 1) entry is 2, so \( \mathrm{B}^{T}\ne\mathrm{B} \).
(C) A is a singular matrix for \( x=\pm\sqrt{\frac{10}{3}} \) — Correct. \( x^2=\frac{10}{3} \) makes |A| = 0.
(D) B is an invertible matrix — Correct. |B| = 7 ≠ 0.