First slide

Applications of determinant

Question

$If [\begin{array}{cc} 1 / 25 & 0 \\ x & 1 / 25 \end{array}] = {[\begin{array}{cc} 5 & 0 \\ - a & 5 \end{array}]}^{- 2}, then the value of x is$

Moderate

A

a/125
B

$2 a / 125$
C

2a/25
D

none of these

Solution

$Let A = [\begin{array}{cc} 5 & 0 \\ - a & 5 \end{array}]$
$\begin{aligned} \Rightarrow adj (A) = [\begin{array}{ll} 5 & 0 \\ a & 5 \end{array}] \\ \begin{array}{ll} \Rightarrow A^{- 1} & = \frac{1}{| A |} [\begin{array}{ll} 5 & 0 \\ a & 5 \end{array}] \\ = \frac{1}{25} [\begin{array}{ll} 5 & 0 \\ a & 5 \end{array}] \\ \Rightarrow A^{- 2} & = {(A^{- 1})}^{2} \\ = \frac{1}{25} [\begin{array}{cc} 5 & 0 \\ a & 5 \end{array}] \frac{1}{25} [\begin{array}{cc} 5 & 0 \\ a & 5 \end{array}] \end{array} \end{aligned}$
$\begin{array}{l} = \frac{1}{625} [\begin{array}{cc} 25 & 0 \\ 10 a & 25 \end{array}] \\ = [\begin{array}{cc} \frac{1}{25} & 0 \\ \frac{2 a}{125} & \frac{1}{25} \end{array}] \\ \begin{array}{l} Now, [\begin{array}{cc} 1 / 25 & 0 \\ x & 1 / 25 \end{array}] = [\begin{array}{cc} \frac{1}{25} & 0 \\ \frac{2 a}{125} & \frac{1}{25} \end{array}] \\ \Rightarrow x = 2 a / 125 \end{array} \end{array}$

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