Question

What is the augmented matrix form of a problem and how do I find it?

Answer 1

An augmented matrix is a matrix formed by appending the columns of two matrices together. It's often used to solve algebra problems. For example, consider:

3x + 2y + 8z = 27
6x + 4y + 4z = 30
x + y + z = 7

The coefficient matrix for this problem is created by taking the coefficients of x in column 1, y in column 2, and z in column 3, and the coefficients of the first equation in row 1, etc:

[3 2 8]
[6 4 4]
[1 1 1]

The solution matrix is a 1-column matrix containing the solutions:

[27]
[30]
[ 7]

The augmented matrix form of the problem is acquired by appending the solution matrix to the coefficient matrix:

[3 2 8 | 27]
[6 4 4 | 30]
[1 1 1 | 7]

We can use this form and elementary row operations to find a solution by reducing the coefficient matrix to an identity matrix:

[3 2 8 | 27]
[2 0 0 | 2] (-4R3)
[1 1 1 | 7]

[3 2 8 | 27]
[1 0 0 | 1] (-0.5R2)
[1 1 1 | 7]

[1 0 6 | 13] (-2R3)
[1 0 0 | 1]
[1 1 1 | 7]

[0 0 6 | 12] (-1R2)
[1 0 0 | 1]
[1 1 1 | 7]

[0 0 1 | 2] (-5R2/6)
[1 0 0 | 1]
[1 1 1 | 7]

[0 0 1 | 2]
[1 0 0 | 1]
[0 1 0 | 4] (-R1 -R2)

[1 0 0 | 1]
[0 1 0 | 4]
[0 0 1 | 2]

So the solution is x = 1, y = 4, z = 2