Question

if A is a matrix such that A^3=I , then show that A is invertible

Answer 1

$\textbf{Concept used:}$

$\text{If A and B are two square matrices of same order, then}$

$\boxed{\bf|AB|=|A|\;|B|}$

$\text{A square matrix A is said to be invertible if |A|\neq\,0 }$

$\textbf{Given:}$

$A^3=I$

$\implies|A^3|=|I|$

$\implies|A|\,|A|\,|A|=|I|$

$\implies|A|^3=1$

$\implies|A|=1$

$\implies\bf|A|\neq0$

$\therefore\textbf{A is invertible}$

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