Let $A = \left[ \begin{array}{ccc}3 & 1 & 2 \\ 1 & 2 & 0 \\ 1 & 1 & 4\end{array} \right]$ and $pC_{11}+4C_{21}-5C_{32} = -2$, where $C_{ij}$ denotes the cofactor of an element $a_{ij}$ of matrix $A$, then the value of $p$ is :

Question

If $\alpha \neq a$, $\beta \neq b$, $\gamma \neq c$ and \[ \begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0,\] then \[ \frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c} \] is equal to:

Answer 1

Step 1: Approach
Compute each minor with its sign using the checkerboard pattern $+,-,+$ in the first row.

Step 2: Cofactors
Delete row 1, column 1: minor $2\cdot4-0\cdot1=8$, sign $+$, so $8$.
Delete row 2, column 1: minor $1\cdot4-2\cdot1=2$, sign $-$, so $-2$.
Delete row 3, column 2: minor $3\cdot0-2\cdot1=-2$, sign $-$, so $+2$.

Step 3: Equation
$8p-8-10=-2$ gives $8p=16$, $p=2$. Option (B).

Final Answer:
The cofactors are 8, -2 and 2, which give p = 2, option (B). \[ \boxed{2} \]

Let \(A = \left[ \begin{array}{ccc}3 & 1 & 2 \\ 1 & 2 & 0 \\ 1 & 1 & 4\end{array} \right]\) and \(pC_{11}+4C_{21}-5C_{32} = -2\), where \(C_{ij}\) denotes the cofactor of an element \(a_{ij}\) of matrix \(A\), then the value of \(p\) is :

Show Hint

The Correct Option is B

Solution and Explanation

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