Q82 / 180 · KCET 2024 · Mathematics

If $A$ is a square matrix, such that $A^2=A$ then $(I+A)^3$ is equal to

  • A
    $A-I$
  • B
    $7 A$
  • Correct answer
    C
    $7 A+I$
  • D
    $I-7 A$

Solution

$(I+A)^3=I^3+A^3+3 I^2 A+3 A^2 I$ $=I+A^3+3 A+3 A^2$ $=I+A^2 \cdot A+3 A+3 A \quad\left[\because A^2=A\right]$ $=I+A \cdot A+6 A$ $=I+A^2+6 A$ $=I+A+6 A \quad\left[\because A^2=A\right]$ $=I+7 A$

Hence, the correct answer is option: 'C'.

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