To solve the system of linear equations using matrices, we can represent the system in the form of a matrix equation \(AX = B\), where:

\[
A = \begin{pmatrix} 3 & -1 \\ 4 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 6 \end{pmatrix}
\]

### Step 1: Write the matrix equation
The system of equations can be written as:

\[
\begin{pmatrix} 3 & -1 \\ 4 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}
\]

### Step 2: Find the inverse of matrix \(A\)
First, calculate the determinant of matrix \(A\):

\[
\text{det}(A) = (3)(1) - (-1)(4) = 3 + 4 = 7
\]

Since the determinant is non-zero, matrix \(A\) is invertible. The inverse of matrix \(A\) is given by:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} 1 & 1 \\ -4 & 3 \end{pmatrix} = \frac{1}{7} \begin{pmatrix} 1 & 1 \\ -4 & 3 \end{pmatrix}
\]

### Step 3: Multiply both sides by the inverse of \(A\)
To solve for \(X\), multiply both sides of the matrix equation by \(A^{-1}\):

\[
X = A^{-1}B = \frac{1}{7} \begin{pmatrix} 1 & 1 \\ -4 & 3 \end{pmatrix} \begin{pmatrix} 1 \\ 6 \end{pmatrix}
\]

### Step 4: Perform the matrix multiplication
Now, calculate the product:

\[
X = \frac{1}{7} \begin{pmatrix} 1 \times 1 + 1 \times 6 \\ -4 \times 1 + 3 \times 6 \end{pmatrix} = \frac{1}{7} \begin{pmatrix} 1 + 6 \\ -4 + 18 \end{pmatrix} = \frac{1}{7} \begin{pmatrix} 7 \\ 14 \end{pmatrix}
\]

### Step 5: Simplify to find the values of \(x\) and \(y\)
Finally, simplify the matrix to get:

\[
X = \begin{pmatrix} \frac{7}{7} \\ \frac{14}{7} \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}
\]

### Final Solution
The solution to the system of equations is:

\[
x = 1, \quad y = 2
\]

Would you like any further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the matrix inverse method change if the determinant was zero?
2. Can you solve a system of equations using the matrix method for larger matrices?
3. How does the matrix method compare to the Gaussian elimination method?
4. What is the significance of the determinant in solving systems of equations using matrices?
5. How would you interpret the solution geometrically using matrices?

**Tip:** When solving systems of equations using matrices, always check if the determinant is non-zero before calculating the inverse, as a zero determinant indicates the matrix is not invertible.

Learn how to solve a system of linear equations using matrices with a step-by-step guide. Includes matrix multiplication and inverse matrix formula applications. Suitable for high school students studying algebra and matrix algebra.

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