Check the statements about a 4x5 matrix with rank 2.

[mathmari]

Hey!

Let $A$ be a $4\times 5$ matrix with rank $2$ and let $U$ be the corresponding row echelon form matrix.

I want to check if the following statements are true or not.

1. If $B$ is a $5\times 5$ invertible matrix, at least two of the columns of $B$ are not in the nulity of $A$.
2. There is a $b\in \mathbb{R}^4$ such that $Ax=b$ has a unique solution.
3. $U$ has exactly one zero row.
4. $U$ has two linearly independent rows.
5. There is a $b\in \mathbb{R}^4$ such that $Ax=b$ has no solution.
6. The equation $Ax=b$ has infinitely many solutions for each $b$.

I have done the following:

1. Since $B$ is a $5\times 5$ invertible matrix, the rank is equal to $5$, or not? From the rank-nullity theorem we get that the nulity is equal to $0$.
   
   What do we get from that? That the statement is wrong?
   
   $ $
2. Since the rank of the matrix is smaller than the number of columns the system $Ax=b$ has more than one solution.
   
   So, the statement is wrong.
   
   $ $
3. Since the rank of $A$ is equal to $2$ the matrix $U$ has $4-2=2$ zero rows.
   
   So, the statement is wrong.
   
   $ $
4. The number of non-zero rows of $U$ is equal to the rank of $A$ and the non-zero rows of $U$ are linearly independent. Therefore, $U$ has $2$ linearly independent rows.
   
   So, the statement is correct.
   
   $ $
5. Similar to (2) this statement is wrong, since we have more than one solution.
   
   $ $
6. Again similar to (2) the statement is correct.

Is everything correct? (Wondering)

 

[I like Serena]

Hey!

Let $A$ be a $4\times 5$ matrix with rank $2$ and let $U$ be the corresponding row echelon form matrix.

I want to check if the following statements are true or not.

1. If $B$ is a $5\times 5$ invertible matrix, at least two of the columns of $B$ are not in the nulity of $A$.
2. There is a $b\in \mathbb{R}^4$ such that $Ax=b$ has a unique solution.
3. $U$ has exactly one zero row.
4. $U$ has two linearly independent rows.
5. There is a $b\in \mathbb{R}^4$ such that $Ax=b$ has no solution.
6. The equation $Ax=b$ has infinitely many solutions for each $b$.

I have done the following:

1. Since $B$ is a $5\times 5$ invertible matrix, the rank is equal to $5$, or not? From the rank-nullity theorem we get that the nulity is equal to $0$.
   
   What do we get from that? That the statement is wrong?

Hey mathmari!

That's about the nullity of $B$. It's not related to the nullity of $A$ is it? (Worried)

Since the rank of $A$ is $2$, its null space is the span of 3 linearly independent vectors.
So 3 columns of $B$ could conceivably fall into the null space of $A$.
But not 4, or could they? (Wondering)

$ $
2. Since the rank of the matrix is smaller than the number of columns the system $Ax=b$ has more than one solution.

So, the statement is wrong.

Couldn't $Ax=b$ have no solution at all? (Wondering)

$ $
3. Since the rank of $A$ is equal to $2$ the matrix $U$ has $4-2=2$ zero rows.

So, the statement is wrong.

$ $
4. The number of non-zero rows of $U$ is equal to the rank of $A$ and the non-zero rows of $U$ are linearly independent. Therefore, $U$ has $2$ linearly independent rows.

So, the statement is correct.

Correct. (Nod)

$ $
5. Similar to (2) this statement is wrong, since we have more than one solution.

$ $
6. Again similar to (2) the statement is correct.

Suppose the row reduced form is:
$$\left(\begin{array}{ccccc|c}1&*&*&*&*&*\\&&1&*&*&*\\&&&&&1\\&&&&&0 \end{array}\right)$$
How many solutions will we have? (Wondering)

 

[mathmari]

That's about the nullity of $B$. It's not related to the nullity of $A$ is it? (Worried)

Since the rank of $A$ is $2$, its null space is the span of 3 linearly independent vectors.
So 3 columns of $B$ could conceivably fall into the null space of $A$.
But not 4, or could they? (Wondering)

Could you explain to me this part further? I got stuck right now. (Wondering)

Couldn't $Ax=b$ have no solution at all? (Wondering)Suppose the row reduced form is:
$$\left(\begin{array}{ccccc|c}1&*&*&*&*&*\\&&1&*&*&*\\&&&&&1\\&&&&&0 \end{array}\right)$$
How many solutions will we have? (Wondering)

So, a system $Ax=b$ has either no solution or infinitely many solution, but it cannot have a unique solution. Is this correct? (Wondering)

 

[I like Serena]

Could you explain to me this part further? I got stuck right now.

Suppose less than 2 columns of $B$ are not in the null space of $A$.
Then at least 4 columns are in the null space.
What is the rank of $AB$ then? (Wondering)

We may want to consider the images of $\mathbf e_1,\ldots,\mathbf e_5$.

So, a system $Ax=b$ has either no solution or infinitely many solution, but it cannot have a unique solution. Is this correct? (Wondering)

Yes. (Nod)

 

[mathmari]

Suppose less than 2 columns of $B$ are not in the null space of $A$.
Then at least 4 columns are in the null space.
What is the rank of $AB$ then? (Wondering)

We may want to consider the images of $\mathbf e_1,\ldots,\mathbf e_5$.

Is the rank of $AB$ less than the minimum of the rank of $A$ and the rank of $B$ ? (Wondering)

Yes. (Nod)

Does it need a proof or can we just say that at the statements (2), (5), (6) and so (2) is false, and (5) and (6) are true? (Wondering)

 

[I like Serena]

Is the rank of $AB$ less than the minimum of the rank of $A$ and the rank of $B$ ?

We have the property that the product of an invertible matrix with another matrix has the same rank as that other matrix.
Can we prove that property, or find it? (Wondering)
And indeed $AB$ would have rank less than $A$, which is then a contradiction.

Does it need a proof or can we just say that at the statements (2), (5), (6) and so (2) is false, and (5) and (6) are true?

In the problem statement you wrote 'I want to check if the following statements are true or not.'
That seems to imply that some checking should be done.
We can use known properties, and I guess we can look them up as well. Otherwise we should verify if they are actually true based on those properties, shouldn't we? (Wondering)
For starters, is (6) really true?

 

[mathmari]

In the problem statement you wrote 'I want to check if the following statements are true or not.'
That seems to imply that some checking should be done.
We can use known properties, and I guess we can look them up as well. Otherwise we should verify if they are actually true based on those properties, shouldn't we? (Wondering)
For starters, is (6) really true?

Ah (6) is not true, because we have for every b. It would be true if we had for some b, or not? (Wondering)

 

[I like Serena]

Ah (6) is not true, because we have for every b. It would be true if we had for some b, or not?

Correct. (Nod)

 

[mathmari]

Correct. (Nod)

So, correct are only the statements (1), (4), (5), or not? (Wondering)

 

[I like Serena]

So, correct are only the statements (1), (4), (5), or not?

Yep. (Nod)