Matrices A and B will be inverse of each other only if - Vidyadip

Question

Matrices A and B will be inverse of each other only if

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Answer

Here it is given that A & B are inverse of each other.
$\therefore$ A^-1 = B ...(i)
Also B^-1 = A ...(ii)
From definition of inverse matrix, we know that-
AA^-1 = I
$\therefore$ A^-1 = B ...from eq (i)
$\Rightarrow$ $ AB = AA^{-1}$ = I...(iii)
Similarly, BB^-1 = I
$\therefore$ B^-1 = A ...(from eq(ii))
$\Rightarrow$ BA = BB⁻¹ = I ...(iv)
So, from (iii) and (iv), we get
AB = BA = I

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