Question

Find the value of a variable $a$ if the cofactors of 1 and -1 are equals to

$\begin{pmatrix} 1 & 2 & -1 \\ 2 & 0 & a \\ 4 & 3 & 5 \end{pmatrix}$.

Answer 1

a = -2

Answer 2

We are given the matrix:

$$A = \begin{pmatrix} 1 & 2 & -1 \\ 2 & 0 & a \\ 4 & 3 & 5 \end{pmatrix}.$$

The problem states that the cofactors corresponding to the entries $1$ (position $(1,1)$) and $-1$ (position $(1,3)$) are equal.

1. Cofactor of $1$ (at $(1,1)$)

   $$C_{11} = (-1)^{1+1} \det\begin{pmatrix} 0 & a \\ 3 & 5 \end{pmatrix} = 1 \times (0\cdot5 - a\cdot3) = -3a.$$

2. Cofactor of $-1$ (at $(1,3)$)

   $$C_{13} = (-1)^{1+3} \det\begin{pmatrix} 2 & 0 \\ 4 & 3 \end{pmatrix} = 1 \times (2\cdot3 - 0\cdot4) = 6.$$

Set the cofactors equal:

$$-3a = 6 \quad \Longrightarrow \quad a = -2.$$

Equate cofactor at (1,1): $-3a$ to that at (1,3): $6$ and solve: $-3a = 6 \Rightarrow a = -2$.