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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?
Solution
Given: A is a square matrix, such that
\[A^2 = A\]
Here,
\[\left( I + A \right)^2 - 3A = \left( I + A \right)\left( I + A \right) - 3A\]
\[ \Rightarrow \left( I + A \right)^2 - 3A = I \times I + I \times A + A \times I + A \times A - 3A \left( \text{using distributive property }\right)\]
\[ \Rightarrow \left( I + A \right)^2 - 3A = I + A + A + A^2 - 3A \left( using I \times I = \text{I and IA} = AI = A \right)\]
\[ \Rightarrow \left( I + A \right)^2 - 3A = I + 2A + A - 3A \left( \because A^2 = A \right)\]
\[ \Rightarrow \left( I + A \right)^2 - 3A = I + 3A - 3A\]
\[ \Rightarrow \left( I + A \right)^2 - 3A = I\]
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