Let `M` be a `3xx3` matrix satisfying `M[0 1 0]=[-1 2 3] ,M[1-1 0]=[1 1-1],a n dM[1 1 1]=[0 0 12]` Then the sum of the diagonal entries of `M` is _________.

Explanation

To find the sum of the diagonal entries of the matrix \( M \), we will use the given conditions step by step.

### Step 1: Define the Matrix
Let \( M \) be a \( 3 \times 3 \) matrix represented as:
\[ M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

### Step 2: Use the First Condition
The first condition given is:
\[ M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \]
Multiplying \( M \) by the vector:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} b \\ e \\ h \end{pmatrix} \]
Setting this equal to the right-hand side:
\[ \begin{pmatrix} b \\ e \\ h \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \]
From this, we can conclude:
\[ b = -1, \quad e = 2, \quad h = 3 \]

### Step 3: Use the Second Condition
The second condition given is:
\[ M \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} \]
Multiplying \( M \) by the vector:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} = \begin{pmatrix} a - b \\ d - e \\ g - h \end{pmatrix} \]
Setting this equal to the right-hand side:
\[ \begin{pmatrix} a - b \\ d - e \\ g - h \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} \]
From this, we can derive:
1. \( a - b = 1 \)
2. \( d - e = 1 \)
3. \( g - h = -1 \)

Substituting \( b = -1 \) and \( e = 2 \):
1. \( a - (-1) = 1 \) → \( a + 1 = 1 \) → \( a = 0 \)
2. \( d - 2 = 1 \) → \( d = 3 \)
3. \( g - 3 = -1 \) → \( g = 2 \)

### Step 4: Use the Third Condition
The third condition given is:
\[ M \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 12 \end{pmatrix} \]
Multiplying \( M \) by the vector:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} a + b + c \\ d + e + f \\ g + h + i \end{pmatrix} \]
Setting this equal to the right-hand side:
\[ \begin{pmatrix} a + b + c \\ d + e + f \\ g + h + i \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 12 \end{pmatrix} \]
From this, we can derive:
1. \( a + b + c = 0 \)
2. \( d + e + f = 0 \)
3. \( g + h + i = 12 \)

Substituting known values:
1. \( 0 - 1 + c = 0 \) → \( c = 1 \)
2. \( 3 + 2 + f = 0 \) → \( f = -5 \)
3. \( 2 + 3 + i = 12 \) → \( i = 7 \)

### Step 5: Summing the Diagonal Entries
Now we have:
- \( a = 0 \)
- \( e = 2 \)
- \( i = 7 \)

The sum of the diagonal entries is:
\[ a + e + i = 0 + 2 + 7 = 9 \]

Thus, the sum of the diagonal entries of \( M \) is \( \boxed{9} \).