If \(A =\begin{bmatrix}x&3\\2&4\end{bmatrix},B=\begin{bmatrix}2&3\\y&3\end{bmatrix}\) and \(C =\begin{bmatrix}z&1\\8&2\end{bmatrix}\) are singular matrices then:
(A) \(x \gt y\)
(B) \(y \gt z\)
(C) \(z \gt x\)
(D) \(x \ne y \ne z\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (C) and (D) only
A singular matrix has determinant zero:
\(|A| = 4x - 6 = 0\)
\( \displaystyle x = \frac{3}{2} \)
\(|B| = 6 - 3y = 0\)
\(y = 2\)
\(|C| = 2z - 8 = 0\)
\(z = 4\)
So \(x = \frac{3}{2}\), \(y = 2\) and \(z = 4\).
Checking the statements
(A) \(x \gt y\) — Incorrect. \(\frac{3}{2} \lt 2\).
(B) \(y \gt z\) — Incorrect. \(2 \lt 4\).
(C) \(z \gt x\) — Correct. \(4 \gt \frac{3}{2}\).
(D) \(x \ne y \ne z\) — Correct. \(\frac{3}{2}\), 2 and 4 are all different.