Question

If \(A = [a_{ij}]_{3\times 3}\) is a matrix such that \(a_{ij} = |2i-5j|\), where \(|.|\) denotes the modulus function, then the element in the \(2^{\text{nd}}\) row and \(3^{\text{rd}}\) column of \(A^{-1}\) is ...

• A. \(3\)
• B. \(1\)
• C. \(0\)
• D. \(-1\)

Hint:

Build the matrix, find its determinant, and use the cofactor C32 for the (2,3) entry of the inverse.

Answer 1

The Correct Option is D

Step 1: Build the matrix
\(a_{ij} = |2i-5j|\) gives \(A = \begin{pmatrix}3 & 8 & 13\\ 1 & 6 & 11\\ 1 & 4 & 9\end{pmatrix}\).

Step 2: Determinant
\[ |A| = 3(54-44) - 8(9-11) + 13(4-6) = 30+16-26 = 20 \]

Step 3: Element of the inverse
The (2,3) element of \(A^{-1}\) equals \(\frac{C_{32}}{|A|}\), where \(C_{32}\) is the cofactor of the element in row 3, column 2.

Step 4: Cofactor
\(M_{32} = \begin{vmatrix}3 & 13\\ 1 & 11\end{vmatrix} = 33-13 = 20\), so \(C_{32} = (-1)^{5}\cdot20 = -20\).

Step 5: Answer
\(\frac{-20}{20} = -1\). Option (D).

Final Answer:
The required element is -1. \[ \boxed{\text{(D)}\ -1} \]