Matrix was given as det `[[x-2,2x-3,3x-4],[2x-3,3x-4,4x-5],[3x-5,5x-8,10x-17]]=Ax^3+Bx^2+Cx+D`. Find the value of `B+C`. is equal to

Options

A. -3
B. 9
C. -1
D. 1

Answer

To solve the given problem, we need to find the determinant of the matrix:

\[ \begin{vmatrix} x - 2 & 2x - 3 & 3x - 4 \\ 2x - 3 & 3x - 4 & 4x - 5 \\ 3x - 5 & 5x - 8 & 10x - 17 \end{vmatrix} \]

and express it in the form \(Ax^3 + Bx^2 + Cx + D\). We will then find the value of \(B + C\).

### Step 1: Calculate the determinant

We will use row operations to simplify the determinant.

1. **Row Operations**:
- Let \(R_2 \rightarrow R_2 - 2R_1\)
- Let \(R_3 \rightarrow R_3 - 3R_1\)

After performing these operations, the matrix becomes:

\[ \begin{vmatrix} x - 2 & 2x - 3 & 3x - 4 \\ 0 & 1 & 2 \\ 0 & 3 & 1 \end{vmatrix} \]

### Step 2: Further simplify the determinant

Now we will perform another row operation:

- Let \(R_3 \rightarrow R_3 - 3R_2\)

This gives us:

\[ \begin{vmatrix} x - 2 & 2x - 3 & 3x - 4 \\ 0 & 1 & 2 \\ 0 & 0 & -5 \end{vmatrix} \]

### Step 3: Calculate the determinant using cofactor expansion

Now we can calculate the determinant using the first row:

\[ \text{det} = (x - 2) \begin{vmatrix} 1 & 2 \\ 0 & -5 \end{vmatrix} \]

Calculating the 2x2 determinant:

\[ = (x - 2)(1 \cdot (-5) - 2 \cdot 0) = -5(x - 2) \]

Thus, we have:

\[ \text{det} = -5(x - 2) = -5x + 10 \]

### Step 4: Express in the form \(Ax^3 + Bx^2 + Cx + D\)

Now we can express the determinant in the required polynomial form:

\[ \text{det} = 0x^3 + 0x^2 - 5x + 10 \]

From this, we can identify:

- \(A = 0\)
- \(B = 0\)
- \(C = -5\)
- \(D = 10\)

### Step 5: Calculate \(B + C\)

Now we need to find \(B + C\):

\[ B + C = 0 + (-5) = -5 \]

### Final Answer

Thus, the value of \(B + C\) is \(-5\).