Find the dependent row of a 4x4 matrix with rank 3.

Question

Consider the matrix $$ A=\begin{pmatrix}1&0&0&-1\\0&1&-1&1\\0&0&-1/3&0\\0&1&0&1\end{pmatrix} $$

I think this matrix has rank 3.

But I cannot identify the row which depends on the other three rows.

Answer 1

Foreword There is no need to use Gauß elimination, but let's do it to see why the rank is three, in another way.

Multiply by $-3$ the third row, and you get $$ \begin{pmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{pmatrix} $$

Now use Gauß elimination doing row $4$ minus row $2$, substituting into row $4$: $$ \begin{pmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} $$

Now it's rather clear that the last two rows are equal, hence the rank is three.

You can see the dependence by reading Gauß elimination backwards, that is: $$\text{row}~3 = \text{row}~ 4 - \text{row} ~ 2$$

Remark

If you don't want to use all this process, you can go to the start and see the dependence as

$$\text{row} ~ 3 = -\frac{1}{3}\left(\text{row} ~ 4 - \text{row} ~ 2\right)$$