Question

If I is the identity matrix and A is a square matrix such that A² = A, then what is the value of (I + A)² = 3A?

Answer 1

Given: A is a square matrix, such that

\[A^2 = A\]

Here,

\[\left( I + A \right)^2 - 3A = \left( I + A \right)\left( I + A \right) - 3A\]

\[ \Rightarrow \left( I + A \right)^2 - 3A = I \times I + I \times A + A \times I + A \times A - 3A \left( \text{using distributive property }\right)\]

\[ \Rightarrow \left( I + A \right)^2 - 3A = I + A + A + A^2 - 3A \left( using I \times I = \text{I and IA} = AI = A \right)\]

\[ \Rightarrow \left( I + A \right)^2 - 3A = I + 2A + A - 3A \left( \because A^2 = A \right)\]

\[ \Rightarrow \left( I + A \right)^2 - 3A = I + 3A - 3A\]

\[ \Rightarrow \left( I + A \right)^2 - 3A = I\]