Proving the determinant of this matrix is $0$: $\left(\begin{smallmatrix}2&1&0&5\\-1&1&1&6\\5&1&-1&4\\5&1&3&0\end{smallmatrix}\right)$

Question

Justify that the determinant of the following matrix is zero $$A=\begin{pmatrix} 2 & 1 & 0 & 5\\ -1 & 1 & 1 & 6\\ 5 & 1 & -1 & 4\\ 5 & 1 & 3 & 0 \end{pmatrix}$$

I have two questions about this example task:

1. If a task says "justify / reason why", am I allowed to do any calculation or am I only allowed to write text?

2. Do you have a better way than mine and can you please show it (I needed to use Gaussian elimination which was very exhausting)?

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The determinant of a square matrix is zero if its row(-vectors) are linearly dependent. Its row(-vectors) are linearly dependent if we get a row with only zeroes and for this we use Gaussian elimination.

Now just apply Gaussian elimination and get a row with only zeroes (I won't do it here to keep it short so just assume we ended up like that).

Would this be a correct way of reasoning or are calculations entirely forbidden for such worded tasks in general? :s

Answer 1

If you find that some row or colum is a linear combination of the others then you can conlude without further calculation otherwise you need to evaluate the determinant or perform RREF.

Sometimes we can also use mixed methods by partial RREF and calculation of the determinant by Laplace method.

In this case we can see that $-2R1+R2+R3=0$ then $$\begin{pmatrix} 2 & 1 & 0 & 5\\ -1 & 1 & 1 & 6\\ 5 & 1 & -1 & 4\\5&1&3&0 \end{pmatrix}\to \begin{pmatrix} 2 & 1 & 0 & 5\\ -1 & 1 & 1 & 6\\ 0 & 0 & 0 & 0\\5&1&3&0 \end{pmatrix}$$

and we can conclude easily that $\det=0$.

Answer 2

You can also see from the future that $\left(2,-19,3,3\right) \in \text{Ker}\left(A\right)$. This implies that $\text{Det}\left(A\right)=0$.

Answer 3

I think the question asks you to find some faster way to proceed than standard Gaussian row reduction, i.e. to find a linear relation between some rows.

For instance, adding the third row to the second yields a new second row which is twice the first.

Answer 4

If a task says "justify / reason why", am I allowed to do any calculation or am I only allowed to write text?

You are allowed to do it in any way you want as long as it's correct and understandable, unless they explicitly ask you to do it one specific way.

Do you have a better way than mine and can you please show it?

Row reduction until one row equals zero is a very reasonable way to do it. Calculating a $4\times4$ determinant by hand is quite a bit of work, so that's probably not better (although it eliminates all guessing). It's possible that reducing column-wise instead of row-wise is shorter in this specific case, but the work to figure out which is shorter is probably not worth it.