Question

State whether the following statement is true or false.
Enter $1$ for true and $0$ for false

If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ then the value of f and g satisfying the matrix equation $A^2+fA+gI=O$ are equal to $-t_r\left(A\right)$ and determinant of A respectively. Given a, b, c, d are non zero reals and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$.

Answer 1

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Here, $tr\left(A\right)=a+d$

and $|A|=ad-bc$

Also, $A^2=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\Rightarrow A^2=\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+dc& bc+d^2\end{array}\right]$
So, $f=-a-d$ and $g=ad-bc$

Consider $A^2+fA+gI$

$=\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+dc& bc+d^2\end{array}\right]+\left[\begin{array}{cc}fa& fb\\ fc& fd\end{array}\right]+\left[\begin{array}{cc}g& 0\\ 0& g\end{array}\right]$
$=\left[\begin{array}{cc}{a}^2+bc+af+g& ab+bd+bf\\ ac+dc+fc& bc+d^2+fd+g\end{array}\right]$
Substituting the values of $f$ and $g$ we get

$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\Rightarrow A^2+fA+gI=O$

Hence, $f$ and $g$ satisfies given matrix equation