If A is a square matrix such that \( \mathrm{A}^{2}=\mathrm{A} \), then \( (\mathrm{I}+\mathrm{A})^{3} \) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \mathrm{I}+7\mathrm{A} \)
Since I commutes with A, the binomial expansion holds:
\( \displaystyle (\mathrm{I}+\mathrm{A})^{3}=\mathrm{I}+3\mathrm{A}+3\mathrm{A}^{2}+\mathrm{A}^{3} \)
Since \( \mathrm{A}^{2}=\mathrm{A} \), also \( \mathrm{A}^{3}=\mathrm{A}^{2}\mathrm{A}=\mathrm{A}^{2}=\mathrm{A} \):
\( \displaystyle (\mathrm{I}+\mathrm{A})^{3}=\mathrm{I}+3\mathrm{A}+3\mathrm{A}+\mathrm{A} \)
\( \displaystyle =\mathrm{I}+7\mathrm{A} \)
NCERT Exemplar:
Matrices
