Question

For a non-singular matrix \(X\), if \(X^2 = I\), then \(X^{-1}\) is equal to:

• A. \(X\)
• B. \(X^2\)
• C. \(I\)
• D. \(O\)

Hint:

If \(X^2 = I\), then \(X\) is said to be its own inverse. This is because the inverse of a matrix \(X\) is the matrix that satisfies \(X \cdot X^{-1} = I\).

Answer 1

The Correct Option is A

We are given that: \[ X^2 = I \] This means: \[ X \cdot X = I \] By the definition of the inverse of a matrix, if: \[ X \cdot X = I \Rightarrow X^{-1} = X \] Since multiplying a matrix by its inverse gives the identity matrix: \[ X \cdot X^{-1} = I \] and here we are told \(X \cdot X = I\), it implies that: \[ X^{-1} = X \] Hence, the inverse of matrix \(X\) is: \[ \boxed{X^{-1} = X} \]