91Ó°ÊÓ

To solve a system of linear equations using matrices, the first step is to write the system in matrix form. This allows us to use matrix algebra to find solutions efficiently.

Given the system:
\[ x - 2y + 3z = 4 \ \ 2x - 4y + 6z = 8 \ \ 3x - 6y + 9z = 14 \]
We can represent it in matrix form as \(AX = B\). Here, matrix \(A\) contains the coefficients of the variables, matrix \(X\) contains the variables, and matrix \(B\) contains the constants on the right-hand side:

\[ A = \begin{pmatrix} 1 & -2 & 3 \ 2 & -4 & 6 \ 3 & -6 & 9 \ \end{pmatrix}, \ \ X = \begin{pmatrix} x \ y \ z \ \end{pmatrix}, \ \ B = \begin{pmatrix} 4 \ 8 \ 14 \ \end{pmatrix} \]

This compact form simplifies the analysis of the system and is essential for applying matrix operations like finding inverses or solving using methods such as Gaussian elimination.

The determinant is a special number that can be calculated from a square matrix. It provides insight into the properties of the matrix, such as whether the matrix is invertible (i.e., it has an inverse).

For the given matrix \(A\):
\[ A = \begin{pmatrix} 1 & -2 & 3 \ 2 & -4 & 6 \ 3 & -6 & 9 \ \end{pmatrix} \]
The determinant of a 3x3 matrix \(A\) is calculated as:
\[ \text{det}(A) = 1 \times (-4 \times 9 - 6 \times -6) - (-2) \times (2 \times 9 - 6 \times 3) + 3 \times (2 \times -6 - (-4) \times 3) \]
Expanding this:
\[ \text{det}(A) = 1 \times ( -36 + 36) - (-2) \times (18 - 18) + 3 \times (-12 + 12) \]
The result simplifies to 0:
\[ \text{det}(A) = 0 \]
This calculation shows that the determinant of matrix \(A\) is zero, which brings us to the next important concept.

A matrix is considered invertible if there exists another matrix that, when multiplied with it, yields the identity matrix. The identity matrix is like the number 1 in matrix algebra: multiplying any matrix by the identity matrix leaves the original matrix unchanged.

For a matrix \(A\) to be invertible, its determinant must be non-zero. From our example, the matrix \(A\) has a determinant of zero:
\[ \text{det}(A) = 0 \]
This means matrix \(A\) is not invertible.

• A matrix with a zero determinant is known as a singular matrix.
• Singular matrices do not have inverses, meaning we cannot use the matrix inverse method to solve the system of equations.

Therefore, since matrix \(A\) in our system is not invertible, the matrix inverse method cannot be used for solving the system of equations. Instead, other methods like Gaussian elimination or analyzing the system for dependencies might be necessary to find solutions.