Question

If A is matrix of order 3x3, then \((A^2)^{-1}\) is equal to

• A. \((-A^2)^2\)
• B. \(A^2\)
• C. \((A^{-1})^2\)
• D. \((-A)^{-2}\)

Answer 1

The Correct Option is C

For a square matrix A, if A has an inverse, denoted as \(A^{-1}\), then \(A \cdot A^{-1} = I\), where I is the identity matrix.
Now, let's evaluate \((A^2)^{-1}\)
\((A^2)^{-1} = (A \cdot A)^{-1}\)
According to the property of matrix inverses, \((A \cdot B)^{-1} = B^{-1} \cdot A^{-1}\) for matrices A and B.
Applying this property to \((A \cdot A)^{-1}\), we get:
\((A \cdot A)^{-1} = A^{-1} \cdot A^{-1}\)
Therefore,\((A^2)^{-1} = A^{-1} \cdot A^{-1}\) (option C).