Question

The inverse of the matrix \(A = \left[ \begin{array}{ccc}2 & -1 & 4 \\ 4 & -3 & 1 \\ 1 & 2 & 1\end{array} \right]\) is \(B = \frac{1}{37}\left[ \begin{array}{ccc}-5 & 9 & 11 \\ -3 & -2 & 14 \\ 11 & -5 & k\end{array} \right]\), then the value of \(k\) is...

• A. \(1\)
• B. \(-1\)
• C. \(2\)
• D. \(-2\)

Hint:

The (3,3) entry of the inverse is the cofactor C33 divided by the determinant.

Answer 1

The Correct Option is D

Step 1: Use AB = I
The entry in row 1, column 3 of $AB$ must be 0. The first row of $A$ is $(2, -1, 4)$ and the third column of $B$ is $\frac{1}{37}(11, 14, k)$.

Step 2: Equation
$2(11) + (-1)(14) + 4k = 0$, so $22 - 14 + 4k = 0$.

Step 3: Solve
$4k = -8$, so $k = -2$.

Step 4: Second check
Row 3 of $A$ is $(1, 2, 1)$. Its product with the third column of $B$ must give 37 after the factor 1/37, so $11 + 28 + k = 37$ gives $k = -2$ again.

Final Answer:
The value of k is -2. This is option (D). \[ \boxed{\text{(D) }-2} \]