Question

How do you solve `x - y + 4z = 6`, `2x + z = 1` and `x + 5y + z = -9` using matrices?

Answer 1

There are several ways to do it, using matrices. I prefer the use of an augmented matrix

Explanation:

The first equation, `x - y + 4z = 6`, makes the following row in the augmented matrix:

`[ (1,-1,4,|,6) ]`

The second equation, `2x + z = 1`, makes the following row in the augmented matrix:

`[ (1,-1,4,|,6), (2,0,1,|,1) ]`

The third equation, `x + 5y + z = -9`, makes the following row in the augmented matrix:

`[ (1,-1,4,|,6), (2,0,1,|,1), (1,5,1,|,-9) ]`

Now, perform elementary row operations until, you obtain an identity matrix on the left.

`R_2-2R_1toR_2`

`[ (1,-1,4,|,6), (0,2,-7,|,-11), (1,5,1,|,-9) ]`

`R_3-R_1toR_3`

`[ (1,-1,4,|,6), (0,2,-7,|,-11), (0,6,-3,|,-15) ]`

`R_3-3R_2toR_3`

`[ (1,-1,4,|,6), (0,2,-7,|,-11), (0,0,18,|,18) ]`

`R_3/18`

`[ (1,-1,4,|,6), (0,2,-7,|,-11), (0,0,1,|,1) ]`

`R_2+7R_3toR_2`

`[ (1,-1,4,|,6), (0,2,0,|,-4), (0,0,1,|,1) ]`

`R_1-4R_2toR_1`

`[ (1,-1,0,|,2), (0,2,0,|,-4), (0,0,1,|,1) ]`

`R_2/2`

`[ (1,-1,0,|,2), (0,1,0,|,-2), (0,0,1,|,1) ]`

`R_1+R_2toR_1`

`[ (1,0,0,|,0), (0,1,0,|,-2), (0,0,1,|,1) ]`

We have an identity matrix on the left, therefore, the solution set is on the right:

`x = 0, y = -2, and z = 1`

Check:

`x - y + 4z = 6`
`2x + z = 1`
`x + 5y + z = -9`

`0 - -2 + 4(1) = 6`
`2(0) + 1 = 1`
`0 + 5(-2) + 1 = -9`

`6 = 6`
`1 = 1`
`-9 = -9`

This checks.