Question

If for the matrix A, \[{A^3} = I\], then find \[{A^{ - 1}}\].
A. \[{A^2}\]
B. \[{A^3}\]
C. A
D. None of these

Answer 1

Hint: We will multiply \[{A^{ - 1}}\] on both sides of the equation and simplify the equation. Then apply the formula of the product of a matrix with its inverse matrix to get the desired solution.

Formula used:
\[A{A^{ - 1}} = I\]

Complete step by step solution:
Given equation is \[{A^3} = I\]
Multiply \[{A^{ - 1}}\] on both sides of the above equation:
\[{A^{ - 1}}{A^3} = {A^{ - 1}}I\]
Rewrite \[{A^3}\] as \[A \cdot {A^2}\]
\[\left( {{A^{ - 1}}A} \right) \cdot {A^2} = {A^{ - 1}}I\]
Now applying the formula \[{A^{ - 1}}A = I\] on the left side expression of the equation:
\[I \cdot {A^2} = {A^{ - 1}}I\] …..(i)
We know that the product of a given matrix with an identity matrix is the given matrix
Thus \[{A^{ - 1}}I = {A^{ - 1}}\] and \[I \cdot {A^2} = {A^2}\]
From equation (i) we get
\[{A^2} = {A^{ - 1}}\]
Hence option A is the correct option.

Additional information:
We can find an inverse matrix of a matrix if the matrix is a square matrix and the determinate value of the matrix must be not equal to zero or the matrix is a non-singular matrix.
The order of an identity matrix that produces by the product of a matrix with its inverse matrix is the same as the order of the given matrix.

Note: Some students do a mistake to find the product of a matrix with an identity matrix. They used the formula \[A \cdot I = I\] which is an incorrect formula. The correct formula is \[A \cdot I = I \cdot A = A\].