Question

If \[ A = \begin{bmatrix} a & b & c \\ d & e & f \\ l & m & n \end{bmatrix} \] is a matrix such that $|A|>0$ and \[ Adj(A) = \begin{bmatrix} 0 & 4 & -6 \\ 10 & 8 & 0 \\ 2 & 4 & -4 \end{bmatrix}, \] then find the value of \[ \frac{cd}{fb} + \frac{ln}{em}. \]

• A. $2a$
• B. $a + m$
• C. $a + b$
• D. $a$

Hint:

Use properties of adjoint and determinant matrices to relate matrix elements and simplify expressions.

Answer 1

The Correct Option is B

Given $|A|>0$, so $A$ is invertible and: \[ A \cdot Adj(A) = |A| I. \] Look at the structure of $Adj(A)$ and use cofactor relations to identify relations between matrix elements. By comparing terms, the given expression simplifies to: \[ \frac{cd}{fb} + \frac{ln}{em} = a + m. \] Hence the answer is $a + m$.