How to determine if this 3x4 Matrix is linearly dependent [closed]

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I have this matrix below:

And i have this question below:

Determines whether the row vectors of A are linearly dependent.

Explain!

Determine the solution quantity for the system of equations

I have tried to search for a solution you can use the determinant but Google says that 3x4 Matrix doesn't have any determinant so i am bit stuck?

And what is solution quantity for the system of equations is it the same as the kernel space?

Thanks in forward!

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3 Answers

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You can get the rank of the matrix, which is by definition

The rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows

So using gaussian elimination you can get the number of linear independent row vectors

$$A = \begin{bmatrix}1&-3&-1&0\\ \:-1&2&0&1\\ \:-1&1&-1&2\end{bmatrix}$$The reduced row echelon form of $A$ is$$\begin{bmatrix}1&0&2&-3\\ 0&1&1&-1\\ 0&0&0&0\end{bmatrix}$$

And since it has one null row you can say that the row vectors a $A$ are not linear independent.

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I took linear algebra a few years ago, but I still might be able to help you out. According to the Wiki page on Linear Independence: "...a set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector."

Thus, to determine that the matrix $A$ is linearly dependant, we need to find a vector $x$ such that $Ax = \vec{0}$. If I am not mistaken, $x = [1,0,1,1]$ is sufficient. I hope this helps!

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Hint:

Add the first row to the second and to the third to obtain an obvious linear relation between rows.

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