Precalculus / Unit 8: Systems and Matrices
8.6 Augmented Matrices and Row Reduction
Notes
An augmented matrix stores a system as one array with a bar before the constants.
- Row operations: swap two rows, multiply a row by a nonzero number, or add a multiple of one row to another.
- Gaussian elimination uses these steps to create zeros below the leading entries.
- Then back-substitute.
- Example: for $x + 3y = 4$ and $2x + 5y = 9$, replace $R_2$ with $R_2 - 2R_1$ to get $-y = 1$.
Practice
Which is NOT a valid row operation?
Adding a constant to the entries is not allowed; you may only add a multiple of another row.
For the system $x + 3y = 4$ and $2x + 5y = 9$, applying $R_2 \to R_2 - 2R_1$ gives the new second row:
$2 - 2(1) = 0$, $5 - 2(3) = -1$, $9 - 2(4) = 1$.
Solve the system $x + 3y = 4$ and $2x + 5y = 9$. What is $x$?
$R_2 - 2R_1$ gives $-y = 1$, so $y = -1$. Then $x + 3(-1) = 4$ gives $x = 7$.
Ready for more? Take the Unit 8 test.