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: Understanding the Question:
We are given a $3 \times 3$ square matrix containing a variable element $x$. The problem states that the multiplicative inverse of this matrix does not exist, and we need to evaluate the exact value of $x$.

Step 2: Key Formula or Approach:
A matrix whose inverse does not exist is called a singular matrix. Mathematically, a matrix $A$ is singular if and only if its determinant is exactly equal to zero: $$ |A| = 0 $$

Step 3: Detailed Explanation:
Let's set the determinant of our given matrix to zero: $$ \begin{vmatrix} 1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6 \end{vmatrix} = 0 $$ Expanding the determinant along the first row: $$ 1 \cdot [(-1)(-6) - 7(4)] - 2 \cdot [4(-6) - 7(2)] + x \cdot [4(4) - (-1)(2)] = 0 $$ Evaluating the minor products within the brackets: $$ 1 \cdot (6 - 28) - 2 \cdot (-24 - 14) + x \cdot (16 + 2) = 0 $$ $$ 1 \cdot (-22) - 2 \cdot (-38) + x \cdot (18) = 0 $$ $$ -22 + 76 + 18x = 0 $$ $$ 54 + 18x = 0 $$ Isolating the variable $x$: $$ 18x = -54 \implies x = -\frac{54}{18} = -3 $$

Step 4: Final Answer:
The value of $x$ for which the matrix inverse does not exist is $-3$, which corresponds to option (A).