Solving Systems of Equations with Matrices

Matrices provide a systematic and powerful method for solving systems of linear equations. This lesson covers expressing systems in matrix form, solving using the inverse matrix, and interpreting cases where the system has no solution or infinitely many solutions.

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Writing Systems in Matrix Form

A system of linear equations:

$$\begin{aligned} a_1 x + b_1 y + c_1 z &= d_1 \\ a_2 x + b_2 y + c_2 z &= d_2 \\ a_3 x + b_3 y + c_3 z &= d_3 \end{aligned}$$a1​x+b1​y+c1​za2​x+b2​y+c2​za3​x+b3​y+c3​z​=d1​=d2​=d3​​

can be written as $A\mathbf{x} = \mathbf{d}$Ax=d where:

$$A = \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}$$A=​a1​a2​a3​​b1​b2​b3​​c1​c2​c3​​​,x=​xyz​​,d=​d1​d2​d3​​​

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Solving Using the Inverse

If $\det(A) \neq 0$det(A)=0, the system has a unique solution:

$\mathbf{x} = A^{-1}\mathbf{d}$x=A−1d

Worked Example 1: Solve:

$$\begin{aligned} 2x + y &= 5 \\ 3x + 2y &= 8 \end{aligned}$$2x+y3x+2y​=5=8​