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...
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...
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The (3,3) entry of the inverse is the cofactor C33 divided by the determinant.
Updated On: Oct 1, 2026
- \(1\)
- \(-1\)
- \(2\)
- \(-2\)
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The Correct Option is D
Solution and Explanation
Step 1: Understand the concept
The inverse is \(A^{-1} = \dfrac{1}{|A|}\text{adj}(A)\), and the adjoint is the transpose of the cofactor matrix. So the entry in position (3,3) is \(C_{33}/|A|\).
Step 2: Find the determinant
\[ |A| = 2(-3 - 2) - (-1)(4 - 1) + 4(8 + 3) = -10 + 3 + 44 = 37 \]
This matches the factor \(\frac{1}{37}\) in \(B\).
Step 3: Find the cofactor \(C_{33}\)
\(C_{33} = (+1)\begin{vmatrix} 2 & -1 \\ 4 & -3 \end{vmatrix} = 2(-3) - (-1)(4) = -6 + 4 = -2\).
Step 4: Result
\(k = -2\), option (D). Check using the first row of \(A\) and the third column of \(B\): \(2 \cdot 11 + (-1)\cdot 14 + 4k = 0\), so \(22 - 14 + 4k = 0\) and \(k = -2\).
Final Answer:
The value of k is -2. This is option (D). \[ \boxed{\text{(D) }-2} \]
The inverse is \(A^{-1} = \dfrac{1}{|A|}\text{adj}(A)\), and the adjoint is the transpose of the cofactor matrix. So the entry in position (3,3) is \(C_{33}/|A|\).
Step 2: Find the determinant
\[ |A| = 2(-3 - 2) - (-1)(4 - 1) + 4(8 + 3) = -10 + 3 + 44 = 37 \]
This matches the factor \(\frac{1}{37}\) in \(B\).
Step 3: Find the cofactor \(C_{33}\)
\(C_{33} = (+1)\begin{vmatrix} 2 & -1 \\ 4 & -3 \end{vmatrix} = 2(-3) - (-1)(4) = -6 + 4 = -2\).
Step 4: Result
\(k = -2\), option (D). Check using the first row of \(A\) and the third column of \(B\): \(2 \cdot 11 + (-1)\cdot 14 + 4k = 0\), so \(22 - 14 + 4k = 0\) and \(k = -2\).
Final Answer:
The value of k is -2. This is option (D). \[ \boxed{\text{(D) }-2} \]
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