Solve the system of equations using matrix method: 2x + 3y + 3z = 5, x - 2y + z = -4, 3x - y - 2z = 3.

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Step-by-Step Solution

Step 1: Represent the system as a matrix equation

First, we represent the given system of linear equations in the matrix form $AX = B$. Here, $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix. We extract the coefficients of $x$, $y$, and $z$ to form matrix $A$, and the constants on the right side of the equations to form matrix $B$.

Step 2: Calculate the determinant of A

To solve for $X$, we need to find the inverse of matrix $A$. The first step in finding the inverse is to calculate the determinant of $A$. If the determinant is non-zero, the inverse exists. We use the formula for the determinant of a $3 \times 3$ matrix.

Step 3: Calculate the adjoint of A

Next, we calculate the adjoint of matrix $A$. The adjoint is the transpose of the cofactor matrix. We find the cofactor for each element, form the cofactor matrix, and then transpose it to get the adjoint.

Step 4: Calculate the inverse of A

Now that we have the determinant and the adjoint of $A$, we can find the inverse of $A$ using the formula $A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$.

Step 5: Solve for X

Finally, we solve for the variable matrix $X$ by multiplying the inverse of $A$ by the constant matrix $B$. This gives us the values for $x$, $y$, and $z$.