KCET · Maths · Matrices
If \(A\) is a square matrix, such that \(A^2=A\) then \((I+A)^3\) is equal to
- A \(A-I\)
- B \(7 A\)
- C \(7 A+I\)
- D \(I-7 A\)
Answer & Solution
Correct Answer
(C) \(7 A+I\)
Step-by-step Solution
Detailed explanation
\((I+A)^3=I^3+A^3+3 I^2 A+3 A^2 I\)
\(=I+A^3+3 A+3 A^2\)
\(=I+A^2 \cdot A+3 A+3 A \quad\left[\because A^2=A\right]\)
\(=I+A \cdot A+6 A\)
\(=I+A^2+6 A\)
\(=I+A+6 A \quad\left[\because A^2=A\right]\)
\(=I+7 A\)
\(=I+A^3+3 A+3 A^2\)
\(=I+A^2 \cdot A+3 A+3 A \quad\left[\because A^2=A\right]\)
\(=I+A \cdot A+6 A\)
\(=I+A^2+6 A\)
\(=I+A+6 A \quad\left[\because A^2=A\right]\)
\(=I+7 A\)
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