Precalculus / Unit 8: Systems and Matrices
8.5 Solving Systems with Matrices
Notes
Write a system as $AX = B$, where $A$ holds the coefficients, $X$ the variables and $B$ the constants.
- If $A$ is invertible, $X = A^{-1}B$.
- Cramer's Rule for two variables: $\displaystyle x = \frac{D_x}{D}$ and $\displaystyle y = \frac{D_y}{D}$, where $D$ is the determinant of the coefficient matrix and $D_x$, $D_y$ replace a column with the constants.
- Example: $2x + y = 7$, $5x + 3y = 18$ has $A^{-1} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$, so $X = \begin{bmatrix} 21 - 18 \\ -35 + 36 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$.
Practice
Which matrix equation represents $x + 2y = 5$ and $3x - y = 1$?
Coefficients go in rows, in the same order as the variables.
Use matrices to solve $2x + y = 7$ and $5x + 3y = 18$. What is $x + y$?
$X = A^{-1}B = (3, 1)$, so $x + y = 4$.
Use Cramer's Rule on $2x + 3y = 13$ and $x - y = -1$. What is $x$?
$D = -2 - 3 = -5$ and $D_x = 13(-1) - 3(-1) = -10$, so $\displaystyle x = \frac{-10}{-5} = 2$.
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