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:
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:
Show Hint
- \( 35 \)
- \( 0 \)
- \( 33 \)
- \( 1 \)
The Correct Option is A
Solution and Explanation
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: \( \boxed{35} \)
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