Question

Let $A=\begin{vmatrix} 5& 5\alpha & \alpha \\\[0.3em] 0 &\alpha &5\alpha \\\[0.3em] 0 &0& 5 \end{vmatrix}$ , If $\left|\,A^2\,\right|=25$,then $\left|\,\alpha\,\right|$ equals

• A. $5^2$
• B. $1$
• C. $\frac{1}{5}$
• D. $5$

Answer 1

Answer: (C)

$\frac{1}{5}$

Answer 2

$| A | =\begin{vmatrix} 5& 5\alpha & \alpha \\\[0.3em] 0 &\alpha &5\alpha \\\[0.3em] 0 &0& 5 \end{vmatrix} = 25 \, \alpha$ Since $|A^2| = 25 \, \therefore \, |A.A| = 25$ $\Rightarrow$ $|A| | A| = 25 \, \Rightarrow (25 \, \alpha)^2 = 25$ $\Rightarrow$ $\alpha^2 = \frac{1}{25} \, \Rightarrow\, |\alpha |= \frac{1}{5} \,$