Matric question based on general system in maths

A typical question involving matrices in mathematics is solving a system of linear equations using matrices. One common method is solving using the inverse of a matrix. Here's an example problem:

**Example Problem:**
Solve the following system of linear equations using matrices:

\[
\begin{aligned}
2x + 3y - z &= 5 \\
4x - y + 5z &= 6 \\
3x + 2y + 4z &= 7
\end{aligned}
\]

### Step 1: Represent the system as a matrix equation.
We can express the system in the form \( A \cdot X = B \), where:
- \( A \) is the coefficient matrix,
- \( X \) is the column matrix of variables,
- \( B \) is the column matrix of constants.

The system can be written as:

\[
\begin{pmatrix}
2 & 3 & -1 \\
4 & -1 & 5 \\
3 & 2 & 4
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
=
\begin{pmatrix}
5 \\
6 \\
7
\end{pmatrix}
\]

Here:
- Coefficient matrix \( A \) is:

\[
A = \begin{pmatrix}
2 & 3 & -1 \\
4 & -1 & 5 \\
3 & 2 & 4
\end{pmatrix}
\]

- Variable matrix \( X \) is:

\[
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
\]

- Constant matrix \( B \) is:

\[
B = \begin{pmatrix}
5 \\
6 \\
7
\end{pmatrix}
\]

### Step 2: Solve for \( X \).
The matrix equation \( A \cdot X = B \) can be solved by multiplying both sides by the inverse of \( A \) (if \( A \) is invertible):

\[
X = A^{-1} \cdot B
\]

We would calculate the inverse of matrix \( A \), and then multiply it by matrix \( B \) to find the values of \( x \), \( y \), and \( z \).

### Step 3: Inverse of Matrix \( A \).
The inverse of a 3x3 matrix \( A \), if it exists, can be found using several methods, such as the adjoint method or Gaussian elimination.

Once the inverse matrix \( A^{-1} \) is determined, the solution for the variables is obtained by:

\[
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
= A^{-1} \cdot B
\]

This provides the values for \( x \), \( y \), and \( z \) that solve the system of equations.

Would you like me to go into detail with the calculation of the inverse matrix or perform it for you?

### Related Questions:
1. How do you determine if a matrix has an inverse?
2. What is the role of the determinant in finding the inverse of a matrix?
3. How can we solve a system of linear equations using Gaussian elimination instead of matrices?
4. What happens if the coefficient matrix is not invertible?
5. Can we solve systems with more than three equations using matrices?

**Tip:** Always check if the determinant of the matrix is non-zero before attempting to find its inverse. A zero determinant means the matrix is singular and has no inverse.

Explore how to solve a system of linear equations using matrices and the inverse matrix method. This problem covers key linear algebra concepts like matrix inversion, the determinant, and applying matrix multiplication. Suitable for Grades 10-12 students.

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