Question:
If \(A^2=A\), then simplify \((I+A)^2 - 7A\), where \(A\) is a square matrix.
If \(A^2=A\), then simplify \((I+A)^2 - 7A\), where \(A\) is a square matrix.
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Expand (I+A)²=I+2A+A², then use A²=A to simplify before subtracting 7A.
Updated On: Sep 23, 2026
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Solution and Explanation
Step 1: Key Formula or Approach:
Expand \((I+A)^2\) using matrix algebra: \((I+A)^2 = I^2 + IA + AI + A^2 = I + 2A + A^2\) (since \(IA=AI=A\) for the identity matrix).
Step 2: Using the given condition \(A^2=A\):
\[ (I+A)^2 = I + 2A + A = I + 3A \]
Step 3: Subtracting \(7A\):
\[ (I+A)^2 - 7A = I + 3A - 7A = I - 4A \]
Final Answer:
The simplified expression is \(I - 4A\). \[ \boxed{I - 4A} \]
Expand \((I+A)^2\) using matrix algebra: \((I+A)^2 = I^2 + IA + AI + A^2 = I + 2A + A^2\) (since \(IA=AI=A\) for the identity matrix).
Step 2: Using the given condition \(A^2=A\):
\[ (I+A)^2 = I + 2A + A = I + 3A \]
Step 3: Subtracting \(7A\):
\[ (I+A)^2 - 7A = I + 3A - 7A = I - 4A \]
Final Answer:
The simplified expression is \(I - 4A\). \[ \boxed{I - 4A} \]
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