If \( A=\begin{bmatrix}\alpha & \beta\\r & \Delta\end{bmatrix} \), such that |A| = 25 , then the value of \( \left|\displaystyle\frac{1}{5}A\right| \) is.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1
For a square matrix of order n, \( |kA|=k^n|A| \), because each of the n rows is multiplied by k. Here n = 2 and \( k=\frac{1}{5} \):
\( \displaystyle \left|\frac{1}{5}A\right|=\left(\frac{1}{5}\right)^2|A| \)
\( \displaystyle =\frac{1}{25}\times 25=1 \)
Why the other options are incorrect
Option (2) \( \displaystyle\frac{1}{5} \) — Incorrect. \( \frac{1}{5} \) is \( \left(\frac{1}{5}\right)^3\times 25 \), which uses the power 3; A is of order 2.
Option (3) 5 — Incorrect. 5 is \( \frac{1}{5}\times 25 \); the scalar must be raised to the power 2, not used once.
Option (4) 125 — Incorrect. 125 is \( 5\times 25 \); the scalar is \( \frac{1}{5} \), not 5, and it must be squared.