If A= [ cc & & - ] is such that A2 = I, then

Question

If $A=\left[\begin{array}{cc} {\alpha} & {\beta} \\ {\gamma} & {-\alpha} \end{array}\right]$ is such that A² = I, then

✓

Answer

In the given question: $A=\left[\begin{array}{ll} {\alpha} & {\beta} \\ {\gamma} & {-\alpha} \end{array}\right]$
Calculating A²: A² = A.A
= $\left[\begin{array}{cc} {\alpha} & {\beta} \\ {\gamma} & {-\alpha} \end{array}\right]\left[\begin{array}{cc} {\alpha} & {\beta} \\ {\gamma} & {-\alpha} \end{array}\right]$
= $\left[\begin{array}{cc} {\alpha \cdot \alpha+\beta \cdot \gamma} & {\alpha \cdot \beta+\beta \cdot(-\alpha)} \\ {\gamma \cdot \alpha+(-\alpha) \cdot \gamma} & {\gamma \cdot \beta+(-\alpha) \cdot(-\alpha)} \end{array}\right]$
= $\left[\begin{array}{cc} {\alpha^{2}+\beta \gamma} & {\alpha \cdot \beta-\alpha \cdot \beta} \\ {\gamma \cdot \alpha-\gamma \cdot \alpha} & {\gamma \cdot \beta+\alpha^{2}} \end{array}\right]$
= $\left[\begin{array}{cc} {\alpha^{2}+\beta \gamma} & {0} \\ {0} & {\beta \gamma+\alpha^{2}} \end{array}\right]$
And given that A² = I
Therefore, $\left[\begin{array}{cc} {\alpha^{2}+\beta \gamma} & {0} \\ {0} & {\beta \gamma+\alpha^{2}} \end{array}\right]=\left[\begin{array}{ll} {1} & {0} \\ {0} & {1} \end{array}\right]$
Comparing corresponding elements we obtain
$\Rightarrow \alpha^{2}+\beta \gamma=1$
$\Rightarrow 1-\alpha^{2}-\beta \gamma=0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks" from Matrices→Matrices — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Matrices — Maths STD 12 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions