If the inverse of `[[1, 2, x], [4, -1, 7], [2, 4, -6]]` does not exist, then x=

Options

A. `3`
B. `-3`
C. `0`
D. `2`

Answer

To find the value of \( x \) for which the inverse of the matrix

\[ A = \begin{bmatrix} 1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6 \end{bmatrix} \]

does not exist, we need to calculate the determinant of the matrix and set it equal to zero, since the inverse of a matrix exists only if its determinant is non-zero.

### Step-by-step Solution:

1. **Write down the determinant formula for a 3x3 matrix:**
The determinant of a 3x3 matrix

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

is calculated as:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

\[ a = 1, b = 2, c = x, d = 4, e = -1, f = 7, g = 2, h = 4, i = -6 \]

2. **Substituting the values into the determinant formula:**

\[ \text{det}(A) = 1((-1)(-6) - (7)(4)) - 2((4)(-6) - (7)(2)) + x((4)(4) - (-1)(2)) \]

3. **Calculate each term:**

- First term:

\[ 1(6 - 28) = 1(-22) = -22 \]

- Second term:

\[ -2(-24 - 14) = -2(-38) = 76 \]

- Third term:

\[ x(16 + 2) = x(18) \]

4. **Combine the terms:**

\[ \text{det}(A) = -22 + 76 + 18x = 54 + 18x \]

5. **Set the determinant equal to zero:**

\[ 54 + 18x = 0 \]

6. **Solve for \( x \):**

\[ 18x = -54 \]
\[ x = \frac{-54}{18} = -3 \]

Thus, the value of \( x \) for which the inverse of the matrix does not exist is

\[ \boxed{-3} \]