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

Answer: (C)

$(A^{-1})^2$

Answer 2

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).