Question

If inverse of $\begin{bmatrix} 1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6 \end{bmatrix}$ does not exist, then $x =$

• A. $-3$
• B. $2$
• C. $3$
• D. $0$

Hint:

Notice a row property dependency: Column 2 elements $(2, -1, 4)$ and Column 1 elements $(1, 4, 2)$ are almost related. Look closely at row 1 and row 3: the first two elements of row 3 $(2, 4)$ are exactly twice the first two elements of row 1 $(1, 2)$. For the entire row 3 to be proportional to row 1 (which forces a zero determinant), the third element must follow: $-6 = 2x \implies x = -3$. A brilliant way to eyeball the answer!

Answer 1

The Correct Option is A

Step 1: Use the singular condition.
No inverse means the matrix is singular, so its determinant is zero.

Step 2: Expand the determinant.
\[ 1(6 - 28) - 2(-24 - 14) + x(16 + 2) = 0 \]
This gives $-22 + 76 + 18x = 0$.

Step 3: Solve.
$54 + 18x = 0$, so $x = -3$.
\[ \boxed{-3,\ \text{option 1}} \]