<s>[INST] Solving system of linear equations involving 3x3 matrix using adjoint matrix 
Solve the following system using the adjoint matrix.
                                    $$2x+4y-10z=-2$$
                                    $$3x+9y-21z=0$$
                                     $$x+5y-12z=1$$

Now, I have tried to solve it, and I got the determinant of matrix $A$ is equal to zero. What is the solution of the above system? Please help me. Thank you.
 [/INST] The system of equation can be expressed in the matrix form as:
$$\begin{pmatrix}2&4&-10\\ 3&9&-21\\1&5&-12\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}-2\\0\\1\end{pmatrix} \iff AX=B.$$
The formula to find $X$:
$$AX=B \iff A^{-1}AX=A^{-1}B \iff X=A^{-1}B=\frac{\text{adj}A}{|A|}B.$$
The determinant of $A$ (performing row operations):
$$|A|=\begin{vmatrix}2&4&-10\\ 3&9&-21\\1&5&-12\end{vmatrix}=\begin{vmatrix}0&-6&14\\ 0&-6&15\\1&5&-12\end{vmatrix}=\begin{vmatrix}0&0&-1\\ 0&-6&15\\1&5&-12\end{vmatrix}=-6.$$
The adjoint of $A$:
$$\text{adj}A=(C_{ij})^{\text{T}}=\begin{pmatrix}9(-12)-(-21)5&-(4(-12)-(-10)5)&4(-21)-(-10)9\\ -(3(-12)-(-21)1)&2(-12)-(-10)1&-(2(-21)-(-10)3)\\3\cdot 5-9\cdot 1&-(2\cdot 5-4\cdot 1)&2\cdot 9-4\cdot 3\end{pmatrix}=\\
=\begin{pmatrix}-3&-2&6\\ 15&-14&12\\6&-6&6\end{pmatrix}$$
Can you calculate $X$?
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