### Determinants

goals
$x + 2y + 3z = 0$
$2x + 3y + 4z = 0$
$7x + 13y + 19z =0$
The system of equations have non trivial solutions
Based on this information answer the questions given below.
Two $n\times n$ square matrices A and B are said to be similar if there exists a non-singular matrix P such that
$P^{-1}A P=B$
Let a, b and c be three real numbers satisfying [a b c] $\begin{bmatrix}1 &9 & 7 \\8 & 2 & 7 \\ 7 & 3 & 7\end{bmatrix} = [0 0 0]$
Find the cofactors of the elements of the following matrices :
(i) $\begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix}$

(ii) $\left[ \begin{matrix} 1 & -1 & 2 \\ -2 & 3 & 5 \\ -2 & 0 & -1 \end{matrix} \right]$
Let system of linear equations $\displaystyle a_{1}x+b_{1}y+c_{1}z=d_{1}a_{2}x+b_{2}y+c_{2}z=d_{2}$ &$\displaystyle a_{3}x+b_{3}y+c_{3}z=d_{3}$ can be expressed in the form $\displaystyle AX=B....(*)$ where $\displaystyle A=\begin{pmatrix}a_{1} &b_{1} &c_{1} \\a_{2} &b_{2} &c_{2} \\a_{3} &b_{3} &c_{3} \end{pmatrix}$ ,$\displaystyle B=\begin{pmatrix}d_{1}\\d_{2}\\d_{3} \end{pmatrix}$ $\displaystyle X= \begin{pmatrix}x\\y\\z\end{pmatrix}$ The above system of equations (*) is said to be consistent with unique solution if A is non singular & the values of the variables x, y, z can be determined by using the equation $\displaystyle X= A^{-1} B$ and if A is singular then system of equations are either consistent with infinitely many solutions or inconsistent with no solution accordingly $\displaystyle (adj A) (B)= 0$ and $\displaystyle (adj A) (B) \neq 0$ where (adj A) is the transpose of cofactor matrix of A Now Assume $\displaystyle A=\begin{pmatrix} a &1 &0 \\1 &b &d \\1 &b &c\end{pmatrix}$,  $\displaystyle B= \begin{pmatrix}a &1 &1 \\0 &d &c \\f &g &h \end{pmatrix}$ $\displaystyle P=\begin{pmatrix}f\\g\\h\end{pmatrix}$,$\displaystyle Q=\begin{pmatrix}a^{2}\\0\\0\end{pmatrix}$,$\displaystyle X=\begin{pmatrix}x\\y\\z\end{pmatrix}$