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Algebra of Matrices — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSAlgebra of Matrices1 Mark
MCQ
If $A$ is a square matrix such that $A^2 = A,$ then $(I + A)^3 - 7A$ is equal to:
  • A
    $A$
  • B
    $I - A$
  • ✓
    $I$
  • D
    $3A$
✓

Answer

Correct option: C.
$I$
Here,
$A^2 = A ...(1)$
$A^3 = A^2A$
$= A^2 [$From $eq. (1)]$
$= A$
$\therefore A^3 = A ...(2)$
We know that $(I + A)^3 = I^3 + 3(I)^2 A + 3(I) A^2 + A^3$
$\Rightarrow (I + A)^3 = I + 3A + 3A + A [$From $eqs. (1)$ and $(2)]$
$\Rightarrow (I + A)^3 = I + 7A$
$\Rightarrow (I + A)^3 - 7A = I$

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