I would like to know if we want to compute the rank of a matrix:
is it sufficient to reduce the matrix to the upper triangular form?
My teacher said we have to reduce to the upper triangular form and count then the number of rows that are not zeros .
But in the below example , it is an upper triangular matrix but the rank is 2 however the number of rows that are not zeros , are 3 !
I think that we have to reduce the matrix to the reduced row echelon form , and it is not sufficient to get an upper triangular!? what is your opinion?
The example
$$A=
\begin{bmatrix}
-4 & 3 & 1\\
0 & 0 & 4\\
0 & 0 &- 3
\end{bmatrix}$$
Best Answer
Your matrix needs to be in "row-echelon form" - you can reduce your matrix further with one more elemenatary row operation . You'll have one zero row, and thus your matrix does not have full rank. "Reduced row echelon form" isn't necessary to determine the rank. So, read up on row-echelon form to understand what it is.
Your last 2 rows are not linearly independent - one row is a scalar-multiple of the other row.
Here are the key points, which are all equivalent:
(1) this matrix does not have full rank
(2) this matrix has determinant = 0
(3) this matrix is not invertible