Question

If the matrix \(\begin{bmatrix}1 & 2 & -1\\-3 & 4 & k\\-4 & 2 & 6\end{bmatrix}\) is singular, then the value of \(k\) is equal to

• A. 3
• B. 4
• C. 5
• D. 6
• E. 7

Hint:

Matrices Tip: Always double-check your signs when expanding determinants! The pattern is always $+ - +$ for the first row, and minus signs inside the $(ad - bc)$ calculation often cause mistakes.

Answer 1

The Correct Option is C

Concept:
A square matrix is called a "singular matrix" if and only if its determinant is equal to zero. To find the unknown variable $k$, we must calculate the determinant of the matrix, set it equal to zero, and solve the resulting algebraic equation.

Step 1: Set the determinant to zero.
Because the matrix is singular, $|A| = 0$: $$\begin{vmatrix}1 & 2 & -1
-3 & 4 & k
-4 & 2 & 6\end{vmatrix} = 0$$

Step 2: Expand along the first row.
Use the standard determinant expansion formula $|A| = a_{11}(C_{11}) - a_{12}(C_{12}) + a_{13}(C_{13})$: $$1 \begin{vmatrix}4 & k
2 & 6\end{vmatrix} - 2 \begin{vmatrix}-3 & k
-4 & 6\end{vmatrix} + (-1) \begin{vmatrix}-3 & 4
-4 & 2\end{vmatrix} = 0$$

Step 3: Evaluate the $2 \times 2$ sub-determinants.
Calculate $ad - bc$ for each $2 \times 2$ matrix: $$1( (4)(6) - (2)(k) ) - 2( (-3)(6) - (-4)(k) ) - 1( (-3)(2) - (-4)(4) ) = 0$$ $$1( 24 - 2k ) - 2( -18 + 4k ) - 1( -6 + 16 ) = 0$$

Step 4: Distribute and simplify the equation.
Expand the terms: $$(24 - 2k) + 36 - 8k - (10) = 0$$ Combine the constant terms and the $k$ terms: $$(24 + 36 - 10) - 2k - 8k = 0$$ $$50 - 10k = 0$$

Step 5: Solve for k.
Isolate $k$ on one side of the equation: $$10k = 50$$ $$k = \frac{50}{10} = 5$$ Hence the correct answer is (C) 5.