Let M be a `3xx3` matrix satisfying
`M[(0),(1),(0)]=[(-1),(2),(3)],M[(1),(-1),(0)]=[(1),(1),(-1)]` and `M[(1),(1),(1)]=[(0),(0),(12)]` then
The sum of the diagonal entries of M is

Options

A. 9
B. 4
C. 1
D. 10

Answer

To solve the problem, we need to find the sum of the diagonal entries of the matrix \( M \) given the conditions provided. Let's denote the matrix \( M \) as follows:

\[ M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

### Step 1: Use the first condition
The first condition states that:

\[ M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \]

This means that the second column of \( M \) must satisfy:

\[ \begin{pmatrix} b \\ e \\ h \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix} \]

From this, we can conclude:
- \( b = -1 \)
- \( e = 2 \)
- \( h = 3 \)

### Step 2: Use the second condition
The second condition states that:

\[ M \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} \]

This gives us the equation:

\[ \begin{pmatrix} a - b \\ d - e \\ g - h \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} \]

Substituting the values of \( b \), \( e \), and \( h \):

1. \( a - (-1) = 1 \) → \( a + 1 = 1 \) → \( a = 0 \)
2. \( d - 2 = 1 \) → \( d = 3 \)
3. \( g - 3 = -1 \) → \( g = 2 \)

### Step 3: Use the third condition
The third condition states that:

\[ M \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 12 \end{pmatrix} \]

This gives us the equation:

\[ \begin{pmatrix} a + b + c \\ d + e + f \\ g + h + i \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 12 \end{pmatrix} \]

Substituting the values we have:

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

### Step 4: Construct the matrix \( M \)
Now we can construct the matrix \( M \):

\[ M = \begin{pmatrix} 0 & -1 & 1 \\ 3 & 2 & -5 \\ 2 & 3 & 7 \end{pmatrix} \]

### Step 5: Calculate the sum of the diagonal entries
The diagonal entries of \( M \) are \( a, e, i \):

\[ \text{Sum of diagonal entries} = a + e + i = 0 + 2 + 7 = 9 \]

### Final Answer
The sum of the diagonal entries of \( M \) is \( 9 \).

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