Question

If A is a square matrix such that A² = A, then (I + A)³ − 7A is equal to

Options

• A

• I-A

• I

• 3A

Answer 1

I

\[Here, \]

\[ A^2 = A . . . \left( 1 \right)\]

\[ A^3 = A^2 A\]

\[ = A^2 \left[ \text{From eq }. \left( 1 \right) \right] \]

\[ = A \]

\[ \therefore A^3 = A . . . \left( 2 \right)\]

\[\text{We know that} \left( I + A \right)^3 = I^3 + 3 \left( I \right)^2 A + 3\left( I \right) A^2 + A^3 \]

\[ \Rightarrow \left( I + A \right)^3 = I + 3A + 3A + A \left[ \text{From eqs }. \left( 1 \right) and \left( 2 \right) \right] \]

\[ \Rightarrow \left( I + A \right)^3 = I + 7A\]

\[ \Rightarrow \left( I + A \right)^3 - 7A = I\]