Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Let $M$ be a $3 \times 3$ matrix satisfying $$M \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}, M \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$$ and $$M \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 12 \end{bmatrix}$$. Then the sum of diagonal entries of $M$ is ____
Step-by-Step Solution
Key Concept: Use the given linear relationships between matrix elements to systematically determine each entry and compute the trace.
Given conditions $a_{12} = -1$, $a_{11} - a_{12} = 0$ yield $a_{11} = 0$. From $a_{11} + a_{12} + a_{13} = 0$ we get $a_{13} = 1$. Using $a_{22} = 2$ and $a_{21} - a_{22} = -1$ gives $a_{21} = 3$. From $a_{21} + a_{22} + a_{23} = 0$ we get $a_{23} = -5$. Finally, $a_{32} = -3$ and $a_{31} - a_{32} = 2$ yield $a_{31} = 2$, and $a_{31} + a_{32} + a_{33} = 12$ gives $a_{33} = 7$. The sum of diagonal elements is $a_{11} + a_{22} + a_{33} = 0 + 2 + 7 = 9$.
Correct Answer: 9