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 matrix-vector equations M·v₁ = w₁, M·v₂ = w₂, M·v₃ = w₃ to extract individual matrix entries by identifying that multiplying M by a standard basis vector or linear combination extracts specific rows or columns. The trace (sum of diagonal entries) equals the coefficient of the matrix when acting on vectors that reveal diagonal elements through the linear system.
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