Question

The product of a $9\times 4$ matrix and a $4\times 9$ matrix contains a variable $x$ in exactly two places. If $D\left(x\right)$ is the determinant of the matrix product such that $D\left(0\right)=1, \, D\left(- 1\right)=1$ and $D\left(2\right)=7$ , then $D\left(- 2\right)$ is equal to

Answer 1

Let, the order of matrix $A$ is $9\times 4$ and matrix $B$ is $4\times 9$
$D\left(x\right)=AB=$ a polynomial of degree $2$ as it has $x$ at $2$ places.
Let, $D\left(x\right)=ax^{2}+bx+c$
$D\left(0\right)=c=1$
$D\left(- 1\right)=a-b+c=1\Rightarrow a-b=0$
$D\left(2\right)=4a+2b+c=7\Rightarrow 4a+2b=6$
$\Rightarrow 2a+b=3$
Solving these two equations, we get,
$a=1,b=1$
$\Rightarrow D\left(x\right)=x^{2}+x+1$
$\Rightarrow D\left(- 2\right)=4-2+1=3$