The Matrix of Equations: Reloaded Examples

Solve this system of equations using row echelon form:

x + 3y + 2z = 3
2x + 8y + 8z = 14
x + 4y + 5z = 10

First, we'll put this in augmented matrix form, then use elementary row operations.

So we have z = 3. Then y + 2(3) = 4, y = -2. Finally, x + 3(-2) + 2(3) = 3, x = 3. Our solution is (3, -2, 3). Pat yourself on the back. You are amazing.

Solve this system of equations using row echelon form:

x + 2y + z = 10
3x + y + 3z = 15
x + z = 4

First, we'll put this in augmented matrix form, then use elementary row operations.

The entire bottom row is zero! That means that 0 = 0, which if you remember back to when we discussed this possibility in Gaussian Elimination, this has infinitely many solutions. This system of equations intersects in an infinite number of spots.

Solve this system of equations using reduced row echelon form:

2y + 6z = 14
x + z = 5
2x + 3z = 11

Put this in augmented matrix form. Get your tools, put your mask on, and start these row operations.

Yes! Reduced echelon form! Our solution is x = 4, y = 4, z = 1 or (4, 4, 1).

Solve using inverse matrices:

We will multiply the right side of the equation by the inverse of the first matrix.

We suggest using the calculator to find the inverse of the matrix and to multiply. Our solution is (1, 2, 5).