Step-by-step Solution:

1. Define the Matrix and the Goal Let the 3 × 3 matrix M be represented as: $M = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ The problem asks for the sum of the diagonal entries of M, which is the trace of M, denoted as Tr(M). Tr(M) = a + e + i.

2. Use the First Given Equation We are given the first condition: $M\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$ Performing the matrix multiplication: $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a(0) + b(1) + c(0) \\ d(0) + e(1) + f(0) \\ g(0) + h(1) + i(0) \end{bmatrix} = \begin{bmatrix} b \\ e \\ h \end{bmatrix}$ By equating this to the given result, we find the second column of M: $\begin{bmatrix} b \\ e \\ h \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$ So, b = -1, e = 2, and h = 3. Note that e = 2 is one of the diagonal entries we need.

3. Use the Second Given Equation The second condition is: $M\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$ Performing the matrix multiplication: $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} a(1) + b(-1) + c(0) \\ d(1) + e(-1) + f(0) \\ g(1) + h(-1) + i(0) \end{bmatrix} = \begin{bmatrix} a - b \\ d - e \\ g - h \end{bmatrix}$ Equating this to the given result: $\begin{bmatrix} a - b \\ d - e \\ g - h \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$ Substitute the known values of b, e, and h:

   • a - (-1) = 1 => a + 1 = 1 => a = 0. This is the first diagonal entry.
   • d - 2 = 1 => d = 3.
   • g - 3 = -1 => g = 2.

4. Use the Third Given Equation The third condition is: $M\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 12 \end{bmatrix}$ Performing the matrix multiplication: $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} a + b + c \\ d + e + f \\ g + h + i \end{bmatrix}$ Equating this to the given result: $\begin{bmatrix} a + b + c \\ d + e + f \\ g + h + i \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 12 \end{bmatrix}$ Substitute all the known values of a, b, d, e, g, h:

   • 0 + (-1) + c = 0 => c = 1.
   • 3 + 2 + f = 0 => f = -5.
   • 2 + 3 + i = 12 => 5 + i = 12 => i = 7. This is the third diagonal entry.

5. Construct the Matrix M and Find the Sum of Diagonal Entries We have now found all the elements of matrix M: $M = \begin{bmatrix} 0 & -1 & 1 \\ 3 & 2 & -5 \\ 2 & 3 & 7 \end{bmatrix}$ The diagonal entries are a = 0, e = 2, and i = 7. The sum of the diagonal entries is: Tr(M) = a + e + i = 0 + 2 + 7 = 9.

Final Answer

The sum of the diagonal entries of M is 9.