Verifying Answers to True/False Questions about Matrices

[Yankel]

Hello

I have been trying to solve a couple of true / false questions, and I am not sure my answers are correct, I would appreciate it if you could verify it.

The first question is:

A and B are matrices such that it is possible to calculate:

\[C=AB+B^{t}A^{t}\]

a. A and B are of the same order
b. C is symmetric
c. BA can be calculated
d. A and B are squared matrices
e. \[ABA^{t}\] can be calculated

My answers are: a - false, b - true , c - true, d - false e - false

The second question is:

A is a 3x3 not invertible matrix:

a. The system Ax=b has a unique solution for every vector b
b. The system Ax=b has infinite number of solutions for every vector b
c. The matrix kA is not invertible for every real number k.
d. If rank(A)=1, then A has at least one row of 0's.
e. There exist a vector b such that Ax=b has infinite number of solutions.

My answers are: a - false, b - false, c - true, d - true, e - true

Is there something wrong with my solutions ?

Thanks a million !

 

[Jameson]

Let's say $A$ is an $(m \times n)$ matrix and $B$ is $(n \times p)$. Then $AB$ is an $(m \times p)$ matrix and and $B^T A^T$ is $(p \times m)$. This implies that $p=m$ for it to be possible to add the two products together to make $C$ but that doesn't mean that $A$ and $B$ are necessarily of the same order. It seems to me that the two matrices could be square but don't have to be.

Assuming my reasoning is correct then your answers for the first one look good. They are all consistent too, which is good.

 

[Opalg]

Hello

I have been trying to solve a couple of true / false questions, and I am not sure my answers are correct, I would appreciate it if you could verify it.

The first question is:

A and B are matrices such that it is possible to calculate:

\[C=AB+B^{t}A^{t}\]

a. A and B are of the same order
b. C is symmetric
c. BA can be calculated
d. A and B are squared matrices
e. \[ABA^{t}\] can be calculated

My answers are: a - false, b - true , c - true, d - false e - false

The second question is:

A is a 3x3 not invertible matrix:

a. The system Ax=b has a unique solution for every vector b
b. The system Ax=b has infinite number of solutions for every vector b
c. The matrix kA is not invertible for every real number k.
d. If rank(A)=1, then A has at least one row of 0's.
e. There exist a vector b such that Ax=b has infinite number of solutions.

My answers are: a - false, b - false, c - true, d - true, e - true

Is there something wrong with my solutions ?

Thanks a million !

I would take a second look at 2d. Everything else looks good.

 

[Yankel]

you are right, my mistake !

If rank(A)=1, the matrix after the elementary row operations has at least one 0 row, not the original A.

Thanks !

 

[blue_raver22]

Hello,

Thank you for reaching out for verification of your answers. I have reviewed your solutions and I have a few comments:

For the first question, your answers for b, c, and e are correct. However, your answer for a is incorrect. A and B do not have to be of the same order for the equation to be valid. For example, A could be a 3x2 matrix and B could be a 2x4 matrix, and the equation would still be valid.

For the second question, your answers for c, d, and e are correct. However, your answer for a is incorrect. A non-invertible matrix does not guarantee a unique solution for every vector b. It is possible for the system to have no solution or infinitely many solutions. Your answer for b is also incorrect for the same reason.

I hope this helps clarify any confusion. Keep up the good work!

 

[blue_raver22]

Hello,

Thank you for reaching out. I am happy to verify your answers to these true/false questions about matrices.

For the first question, your answers are mostly correct. However, for option d, A and B do not necessarily have to be squared matrices. As long as they have compatible dimensions, the operation can be performed. Therefore, the correct answer would be false.

For the second question, your answers are also mostly correct. However, for option e, it is not always true that there exists a vector b such that Ax=b has an infinite number of solutions. This statement is only true if A is a singular matrix (determinant = 0). Therefore, the correct answer would be false.

Overall, your solutions are mostly correct. Keep up the good work! If you have any further questions or need clarification, please don't hesitate to ask.

Best,