Question

Solving linear equation with matrices

So, we were taught how to solve linear equations using the Gauss-Jordan algorithm. What bothers me is that I can't guess when the matrix has been reduced as much as possible. Is there some way to predict it?

I can give an example from the problems I've been solving if it will be more clear.

Answer 1

Have you learnt the reduced row-echelon form of a matrix?

Answer 2

@RolyPoly Yes, that's the part where I'm not sure when to 'stop'. I mean, I perform operations to make sure there are as much zeroes under the main diagonal as possible but then, I sometimes leave the matrix earlier while the solved examples proceed further. I'll write down one of them and attach a picture to make it more clear.

Answer 3

It would be better if you can write an example here :)

Answer 4

Here it is. The part in red is the final answer that I was supposed to get. I can see how to get to there but I don't know what's hinting that I should subtract and divide further from my answer.

Answer 5

Is this the matrix?
\[\left[\begin{matrix}0 & 0 & 6 & 2 &-4 & -8 \\ 0 & 0 & 3 & 1 & -2 & -4 \\ 0&-3& 1 & 4 & -9 & 1 \\ 6 & -9 & 0 & 11 & 19 & -3\end{matrix}\right]\]

Answer 6

Yes. It was given as a system of equations but on the paper I only wrote the matrix of coefficients in front of x, A, and the row vector b of the right-hand side coefficients.

Answer 7

Is the 3,1th entry 0?

Answer 8

In my answer?

Answer 9

I'm sorry that I am having a hard time reading the entries of matrix A. Can you please type the matrix A like this:
Row 1 : x x x x x x x , where x are the numbers (entries) in row 1
Row 2 : x x x x x x x
And so on...

Answer 10

so:
R1: 0 0 6 2 -4 -8
R2: 0 0 3 1 -2 -4
R3: 2 -3 1 4 -7 1
R4: 6 -9 0 11 -19 3

Answer 11

Not sure if you can see it clearly.

Answer 12

For every row, the first non-zero entry must be 1, so for the first row, you need to divide the whole row by 2. Similarly, for the second row, you need to divide the whole row by 3. I bet you know what to do with the 4th row then. :)
Then, the leading 1 is the ONLY non-zero entry in the corresponding column, so you need to use type III ERO to make it the only non-zero entry in the column.

Answer 13

Please pay attention to the following definitions:
1) A matrix is said to be in row-echelon form if it satisfies the following conditions:
i) zero rows, if any, must be at the bottom of the matrix
ii) the first non-zero entry of a non-zero row must be 1, called a leading 1
iii) a leading 1 must be on the right of the leading 1 in the rows above
2) It is said to be in reduced row echelon form if it furthermore satisfies the following:
i) any leading 1 must be the only non-zero entry in the column.

Answer 14

Oh, right, so that's the part I haven't got right - that the first non-zero entry has to be 1. Thank you so much for the detailed explanations! It's all clear now.

Answer 15

You're welcome. :)