Question

The element in the third row and the second column in the inverse matrix of a matrix \(\left[ \begin{array}{ccc}1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 4\end{array} \right]\) is

• A. \(\frac{3}{2}\)
• B. \(\frac{2}{3}\)
• C. \(\frac{-3}{2}\)
• D. \(\frac{-2}{3}\)

Hint:

The (3,2) entry of the inverse is the cofactor C23 divided by the determinant.

Answer 1

The Correct Option is A

Step 1: Formula
$(A^{-1})_{ij} = C_{ji}/|A|$, so for row 3 column 2 we need $C_{23}$.

Step 2: Numbers
$|A| = 4$ and $C_{23} = -\begin{vmatrix} 1 & 3 \\ 3 & 3 \end{vmatrix} = -(-6) = 6$.

Step 3: Result
$6/4 = 3/2$. Option (A).

Final Answer:
Option (A). \[ \boxed{\frac{3}{2}} \]