Determining value of r that makes the matrix linearly dependent

[Sunwoo Bae]

Homework Statement

Given in the context

Relevant Equations

A matrix is linearly dependent if:
1. There are more vectors than entries
2. The matrix contains 0 vector
3. one vector is multiple of another

for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection.

As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon form in order to check if there are any free variables, as presence of free variable indicates the system is linearly dependent. Through reducing the matrix, I can find out that value of 4 would make the last row of the matrix all zero, making x3 a free variable. Therefore, I can conclude that r=4.

The following is my work:

Are my answers and reasoning valid for these two problems? Also, is my approach to solving these problems correct?

Thank you for your help.

 

[Orodruin]

The most straightforward way of telling whether the columns of a square matrix are linearly independent is to take the determinant. If zero, the columns are linearly dependent.

That said, your reduction does seem correct and results in the correct answer.

 

[PeroK]

for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection.

I guess you mean linearly dependent here?

 

[PeroK]

PS for part b) I would have solved the equations: $$a + 2b = -1, \ \ 2a + b = 7$$ then used $a$ and $b$ to determine $r$.

 

[Sunwoo Bae]

I guess you mean linearly dependent here?

yes. That was a typo. sorry