Solve the system of equations: 2x + 3y - z = 5, x - y + 2z = 1, 3x + 2y + z = 8 using matrix method.

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Step-by-Step Solution

Step 1: Represent the system as a matrix equation

We can represent the given system of linear equations in the matrix form AX = B$, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. This is the first step in solving the system using the matrix method.

Step 2: Calculate the determinant of A

Next, we calculate the determinant of the coefficient matrix A$. If the determinant is zero, the system either has no solution or infinitely many solutions. In this case, \\det(A) = 0$, which means the system is either inconsistent or dependent.

Step 3: Calculate the adjoint of A

To further analyze the system, we need to calculate the adjoint of matrix A$. The adjoint matrix is the transpose of the cofactor matrix. We find each cofactor C_{ij} and then arrange them to form the adjoint.

Step 4: Calculate (adj A)B

Now, we multiply the adjoint of A by the constant matrix B$. This product helps us determine the nature of the solutions. If (\text{adj } A)B is not a zero matrix when \\det(A) = 0$, the system is inconsistent.