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Maths-

General

Easy

Question

If for a matrix A, A³ = I then A^–1 equals -

A²
A
A³
None of these

hint Hint:

The inverse of matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A^-1, and A · A^-1= A^-1· A = I, where I is the identity matrix. The matrix whose determinant is non-zero and for which the inverse matrix can be calculated is called an invertible matrix.

The correct answer is: A²

Given, $A cubed equals I$
$A cubed equals I$

Multiplying both sides by $A to the power of negative 1 end exponent$ , we get
$rightwards double arrow A to the power of 3 space end exponent space A to the power of negative 1 end exponent equals I space A to the power of negative 1 end exponent A to the power of 3 space end exponent space c a n space b e space s p l i t space i n t o space A squared space. space space A rightwards double arrow A squared left parenthesis A space A to the power of negative 1 end exponent right parenthesis space equals space I space A to the power of negative 1 end exponent W e space k n o w space t h a t space A A to the power of negative 1 end exponent space equals space I space a n d space I A to the power of negative 1 end exponent space equals space space A to the power of negative 1 end exponent space. space S u b s t i t u t i n g space t h e s e space v a l u e s space w e space g e t rightwards double arrow A squared left parenthesis I right parenthesis equals space A to the power of negative 1 end exponent rightwards double arrow A squared equals space A to the power of negative 1 end exponent T h u s comma space i f space f o r space a space m a t r i x space A comma space A 3 space equals space I space t h e n space A – 1 space e q u a l s space minus space A squared$

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