Solve System of Linear Equations using Matrices

Math Problem Statement

solve with help of matrices

To solve the system of linear equations using matrices, we can rewrite the given equations in matrix form. The provided system is:

$\frac{x - 3y}{2} = 0 \quad \text{and} \quad 2x + y = 7$

First, we rewrite the first equation to eliminate the fraction:

$x - 3y = 0$

Now, we have the system:

x - 3y = 0 \\ 2x + y = 7 \end{cases}$$ Next, we express this system in matrix form $$AX = B$$: $$\begin{pmatrix} 1 & -3 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 7 \end{pmatrix}$$ Here, the matrix $$A$$ is the coefficients matrix, $$X$$ is the variables matrix, and $$B$$ is the constants matrix. We need to solve for $$X$$ using the equation $$X = A^{-1}B$$. 1. Calculate the determinant of $$A$$: $$\text{det}(A) = (1)(1) - (-3)(2) = 1 + 6 = 7$$ Since the determinant is non-zero, $$A$$ is invertible. 2. Find the inverse of $$A$$: \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} 1 & 3 \\

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Linear Equations
Matrices
Determinants
Matrix Inversion

Matrix form of linear equations
Determinant calculation
Matrix inversion formula

Invertibility of matrices

Grades 10-12

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