QJEEM21Ma4d13: JEE Main 2021 Mathematics - Class 12 JEEM21Ma4d13 Solution: The number of elements in the set { A = ( a b 0 d ) : a , b , d …

Question

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The number of elements in the set

{A = (\begin{matrix} a & b \\ 0 & d \end{matrix}) : a, b, d \in {- 1, 0, 1} a n d {(I - A)}^{3} = I - A^{3}}

, where I is 2

\times

2 identity matrix, is :

Accepted Answer

{(I - A)^3} = {I^3} - {A^3} - 3A(I - A) = I - {A^3}

\Rightarrow 3A(I - A) = 0

or

{A^2} = A

\Rightarrow [\begin{matrix} a^{2} & a b + b d \\ 0 & d^{2} \end{matrix}] = [\begin{matrix} a & b \\ 0 & d \end{matrix}]

\Rightarrow {a^2} = a,b(a + d - 1) = 0,{d^2} = d

If b

\ne

0, a + d = 1

\Rightarrow

4 ways

If b = 0, a = 0, 1 & d = 0, 1

\Rightarrow

4 ways

\Rightarrow

Total 8 matrices

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