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,
$$\begin{gathered} {\left(I+A\right)}^2-3A=\left(I+A\right)\left(I+A\right)-3A \\ \Rightarrow {\left(I+A\right)}^2-3A=I\times I+I\times A+A\times I+A\times A-3A\left(\mathrm{using}\mathrm{distributive}\mathrm{property}\right) \\ \Rightarrow {\left(I+A\right)}^2-3A=I+A+A+A^2-3A\left(\mathrm{using}I\times I=I\mathrm{and}IA=AI=A\right) \\ \Rightarrow {\left(I+A\right)}^2-3A=I+2A+A-3A\left(\because A^2=A\right) \\ \Rightarrow {\left(I+A\right)}^2-3A=I+3A-3A \\ \Rightarrow {\left(I+A\right)}^2-3A=I \end{gathered}$$