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).

Answer 2

We are given that \( A \) is a matrix of order \( 3 \times 3 \). We are to find: \[ (A^2)^{-1} \] Recall the property of matrix inverses: \[ (AB)^{-1} = B^{-1}A^{-1} \] and in particular, \[ (A^2)^{-1} = (A \cdot A)^{-1} = A^{-1} \cdot A^{-1} = (A^{-1})^2 \] Therefore, \[ (A^2)^{-1} = (A^{-1})^2 \] Correct answer: \((A^{-1})^2\) 

Answer 3

We are given a matrix A of order 3x3 and asked to find the expression for \((A^2)^{-1}\).

We know that for any invertible matrix B, \((B^{-1})^{-1} = B\).

Also, \((AB)^{-1} = B^{-1}A^{-1}\), where A and B are invertible matrices.

Using these properties, we have:

\((A^2)^{-1} = (AA)^{-1}\)

\((A^2)^{-1} = A^{-1}A^{-1}\)

\((A^2)^{-1} = (A^{-1})^2\)

Now let's check if any of the other options are equivalent to \((A^{-1})^2\):

• \((-A^2)^2 = (-A^2)(-A^2) = A^4\). This is not equal to \((A^{-1})^2\).
• \(A^2\) is simply \(A*A\), which is not equal to \((A^{-1})^2\).
• \((-A)^{-2} = ((-A)^{-1})^2 = ((-1)A^{-1})^2 = (-1)^2(A^{-1})^2 = (A^{-1})^2\).

Thus, \((A^2)^{-1} = (A^{-1})^2 = (-A)^{-2}\)

The correct answer is \( (A^{-1})^2\)

Answer:

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