Matrix inverse: Solve each system of equations by using the inverse of the coeffcient matrix if it exists and by the Gauss-Jordan method if the   inverse doesn’t exist. 
2x + y =1
3y + z =8
4x - y -3z =8

We are given the system of equations:

1. \( 2x + y = 1 \)
2. \( 3y + z = 8 \)
3. \( 4x - y - 3z = 8 \)

### Step 1: Write the system as a matrix equation
This system of equations can be expressed in matrix form as:

\[
A \mathbf{x} = \mathbf{b}
\]

Where:

\[
A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 4 & -1 & -3 \end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix} 1 \\ 8 \\ 8 \end{pmatrix}
\]

### Step 2: Check if the inverse of the matrix \( A \) exists
To determine whether the inverse of \( A \) exists, we first calculate its determinant. If the determinant is non-zero, the inverse exists.

The determinant of a 3x3 matrix \( A \) is given by:

\[
\det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
\]

For matrix \( A \):

\[
A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 4 & -1 & -3 \end{pmatrix}
\]

Let me calculate the determinant.The determinant of matrix \( A \) is \( -12 \), which is non-zero. Therefore, the inverse of \( A \) exists, and we can solve the system using the inverse of the coefficient matrix.

### Step 3: Solve the system using the inverse of \( A \)

The solution to the system is given by:

\[
\mathbf{x} = A^{-1} \mathbf{b}
\]

Let me calculate the inverse of \( A \) and use it to find \( \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \).The solution to the system of equations using the inverse of the coefficient matrix is:

\[
x = -2, \quad y = 5, \quad z = -7
\]

### Summary:
- \( x = -2 \)
- \( y = 5 \)
- \( z = -7 \)

Would you like further details on the inverse matrix or Gauss-Jordan method?

Here are 5 related questions to deepen your understanding:
1. How do we calculate the inverse of a matrix manually?
2. What is the Gauss-Jordan elimination method?
3. How do we interpret the determinant of a matrix geometrically?
4. Can every matrix be inverted? What are the conditions?
5. What is the relationship between the determinant and matrix singularity?

**Tip:** Always check the determinant before attempting to compute the inverse of a matrix.

Matrix inverse: Solve each system of equations by using the inverse of the coefficient matrix if it exists and by the Gauss-Jordan method if the inverse doesn’t exist. 2x + y = 1, 3y + z = 8, 4x - y - 3z = 8

Learn how to solve a system of equations using the matrix inverse and Gauss-Jordan elimination. This problem covers key concepts in linear algebra, including matrix inversion and determinant calculation, and is suitable for undergraduate students.

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