Explain rank of matrix

Question

$$ A = \left[\begin{array}{cccc}4& 1& 5& 2\\ 1& 2& 3& 4\\ 2& 0& 2& 0\\ 3& 4& 80& 22\end{array}\right]$$

Can anyone please explain why rank is $3$?

Answer 1

When you use gauss-jordan elimination on A, you get $$ A= \begin{bmatrix} 1 & 0 & 0 & -14/73 \\ 0 & 1 & 0 & 132/73 \\ 0 & 0 & 1 & 14/73\\ 0 & 0 & 0 & 0\\ \end{bmatrix} $$

And because the rank A = # nonzero pivots of rref(A) [rref = reduce row echelon form], the rank is 3.

There is more that goes into it, but that's how you can find the rank of A.

Answer 2

Try putting it in row reduced echelon form. You will get a row of all zeroes which means the zero row is some linear combination on the other nonzero rows.

The number of nonzero rows is equal to the rank of the matrix.