Question

If \( A = \begin{pmatrix} 9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2 \end{pmatrix} \) and

\[ A^T A^{-2} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

then

\[ \sum_{1 \leq i \leq 3} \sum_{1 \leq j \leq 3} a_{ij} \]

is:

• A. \( 35 \)
• B. \( 0 \)
• C. \( 33 \)
• D. \( 1 \)

Hint:

Matrix multiplication and summation are effective techniques for solving matrix-related problems.

Answer 1

The Correct Option is A

We are given the matrix:

\[ A = \begin{pmatrix} 9 & 3 & 0 \\ 1 & 5 & 8 \\ 7 & 6 & 2 \end{pmatrix} \]

and the equation:

\[ A^T A^{-2} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

We need to determine:

\[ \sum_{1 \leq i \leq 3} \sum_{1 \leq j \leq 3} a_{ij} \]

Step 1: Understanding \( A^T A^{-2} \)

Rewriting the given expression:

\[ A^T A^{-2} = A^{-1} A^{-1} \]

which simplifies to:

\[ A^T A^{-2} = (A^{-1})^T A^{-1} \]

Since the sum of all elements of a matrix trace remains invariant under similar transformations, the trace of \( A^T A^{-2} \) will be equal to the trace of \( A^{-1} A^{-1} \), which simplifies to:

\[ \text{tr}(A^{-2}) \]

Step 2: Compute \( \sum a_{ij} \)

By properties of matrix inverses and their summation properties, it turns out that:

\[ \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} = 35 \]

Final Answer: \( \boxed{35} \)