2. Consider the system of linear equations
\[
\begin{array}{r}
2 y+3 z=0 \\
x+2 y+z=0 \\
-2 x-2 y+z=0
\end{array}
\]
b) $[6$ marks] Write the system in matrix form $A X=B$. Find $\operatorname{det} A$ by cofactor expansion.
c) [2 marks] Is the system consistent? Is the system homogeneous?

Step 1 ：The system of linear equations can be written in matrix form as follows: \[\begin{array}{ccc|c} 0 & 2 & 3 & 0 \\ 1 & 2 & 1 & 0 \\ -2 & -2 & 1 & 0 \end{array}\]

Step 2 ：This can be represented as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Step 3 ：The determinant of A can be found using the formula: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg) where a, b, c are the elements of the first row of the matrix, d, e, f are the elements of the second row, and g, h, i are the elements of the third row.

Step 4 ：The system is consistent if the determinant of A is not equal to zero. If the determinant is zero, the system is inconsistent.

Step 5 ：The system is homogeneous if the constant matrix B is a zero matrix. In this case, B is a zero matrix, so the system is homogeneous.

Step 6 ：The determinant of the matrix A is \(\boxed{0}\). The system is \(\boxed{inconsistent}\) and \(\boxed{not homogeneous}\).