Question

if A is a square matrix of order 3 and if A^-1 = A^k then the value of k is

Answer 1

$\textbf{Concept used:}$

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

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

$\textbf{Given:}$

$\text{A is a square matrix of order 3 with A^{-1}=A^k }$

$\textbf{To find : value of k}$ 

$A^{-1}=A^k$

$\implies|A^{-1}|=|A^k|$

$\implies|A^{-1}|=|A.A.A.....\text{k times}|$

$\implies\frac{1}{|A|}=|A|\;|A|\;|A|.....\text{k times}$

$\implies\;|A|^{-1}=|A|^k$

$\text{Equating powers on both sides, we get}$

$\bf\;k=-1$

$\therefore\textbf{The value of k is -1}$

Find more:

1.If A is a non-singular square matrix of order 3 such that A^2 = 3A, then

value of det.A is

(A) -3

(B) 3IA|

(C) 9

D) 27​

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2.A is a square matrix with |A|=4 then find the value of |A. (adjA)|

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

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