Question
This question is meant to serve as an answer to the following: If you sum two rank-1 matrices, do you get a rank-2 matrix? What about products? More generally, what rank is the sum of a rank-r_1 and a rank-r_2 matrix?"
Let \mathbf{A}= \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix}, \mathbf{B}= \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} and \mathbf{C}= \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} and \mathbf{D}= \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}. Observe that all four of these matrices are rank 1.
There are many ways to determine rank. Here is one useful fact that you could use for this problem:
“Every rank-1 matrix can be written as an outer product. Conversely, every outer product \mathbf{u}\mathbf{v}^ T is a rank-1 matrix."
For example, \mathbf{A}= \mathbf{u}\mathbf{v}^ T, \mathbf{B}= \mathbf{v}\mathbf{v}^ T, \mathbf{C}= \mathbf{w}\mathbf{w}^ T and \mathbf{D}= \mathbf{x}\mathbf{x}^ T, where
\mathbf{u}= \begin{pmatrix} 1 \\ 3 \end{pmatrix}, \mathbf{v}= \begin{pmatrix} -1 \\ 1 \end{pmatrix}, \mathbf{w}= \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \mathbf{x}= \begin{pmatrix} 1 \\ 1 \end{pmatrix}.
Which combination of these matrices has rank 2? Choose all that apply.
\mathbf{A}+\mathbf{A}
\mathbf{A}+\mathbf{B}
\mathbf{A}+\mathbf{C}
\mathbf{A}\mathbf{B}
\mathbf{A}\mathbf{C}
\mathbf{B}\mathbf{D}
incorrect
Which combination of these matrices has rank 1? Choose all that apply.
\mathbf{A}+\mathbf{A}
\mathbf{A}+\mathbf{B}
\mathbf{A}+\mathbf{C}
\mathbf{A}\mathbf{B}
\mathbf{A}\mathbf{C}
\mathbf{B}\mathbf{D}
Let \mathbf{A}= \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix}, \mathbf{B}= \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} and \mathbf{C}= \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} and \mathbf{D}= \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}. Observe that all four of these matrices are rank 1.
There are many ways to determine rank. Here is one useful fact that you could use for this problem:
“Every rank-1 matrix can be written as an outer product. Conversely, every outer product \mathbf{u}\mathbf{v}^ T is a rank-1 matrix."
For example, \mathbf{A}= \mathbf{u}\mathbf{v}^ T, \mathbf{B}= \mathbf{v}\mathbf{v}^ T, \mathbf{C}= \mathbf{w}\mathbf{w}^ T and \mathbf{D}= \mathbf{x}\mathbf{x}^ T, where
\mathbf{u}= \begin{pmatrix} 1 \\ 3 \end{pmatrix}, \mathbf{v}= \begin{pmatrix} -1 \\ 1 \end{pmatrix}, \mathbf{w}= \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \mathbf{x}= \begin{pmatrix} 1 \\ 1 \end{pmatrix}.
Which combination of these matrices has rank 2? Choose all that apply.
\mathbf{A}+\mathbf{A}
\mathbf{A}+\mathbf{B}
\mathbf{A}+\mathbf{C}
\mathbf{A}\mathbf{B}
\mathbf{A}\mathbf{C}
\mathbf{B}\mathbf{D}
incorrect
Which combination of these matrices has rank 1? Choose all that apply.
\mathbf{A}+\mathbf{A}
\mathbf{A}+\mathbf{B}
\mathbf{A}+\mathbf{C}
\mathbf{A}\mathbf{B}
\mathbf{A}\mathbf{C}
\mathbf{B}\mathbf{D}
Answers
GPT 3.5
For a matrix to have a rank of 2, it must require at least two linearly independent vectors as columns or rows.
The sum of two rank-1 matrices will not necessarily result in a rank-2 matrix. Let's check each combination:
1) \mathbf{A}+\mathbf{A}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} + \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} = \begin{pmatrix} -2 & 2 \\ -6 & 6\end{pmatrix}. The rank of this matrix is still 1.
2) \mathbf{A}+\mathbf{B}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} + \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} = \begin{pmatrix} 0 & 0 \\ -4 & 4\end{pmatrix}. The rank of this matrix is still 1.
3) \mathbf{A}+\mathbf{C}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} -1 & 1 \\ -3 & 4\end{pmatrix}. The rank of this matrix is still 1.
4) \mathbf{A}\mathbf{B}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} = \begin{pmatrix} -2 & 2 \\ -6 & 6\end{pmatrix}. The rank of this matrix is still 1.
5) \mathbf{A}\mathbf{C}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 3\end{pmatrix}. The rank of this matrix is still 1.
6) \mathbf{B}\mathbf{D}:
This results in the matrix \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}. The rank of this matrix is 0.
So, the only combination that results in a rank-2 matrix is:
\mathbf{B}\mathbf{D}
The sum of two rank-1 matrices will not necessarily result in a rank-2 matrix. Let's check each combination:
1) \mathbf{A}+\mathbf{A}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} + \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} = \begin{pmatrix} -2 & 2 \\ -6 & 6\end{pmatrix}. The rank of this matrix is still 1.
2) \mathbf{A}+\mathbf{B}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} + \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} = \begin{pmatrix} 0 & 0 \\ -4 & 4\end{pmatrix}. The rank of this matrix is still 1.
3) \mathbf{A}+\mathbf{C}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} -1 & 1 \\ -3 & 4\end{pmatrix}. The rank of this matrix is still 1.
4) \mathbf{A}\mathbf{B}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} = \begin{pmatrix} -2 & 2 \\ -6 & 6\end{pmatrix}. The rank of this matrix is still 1.
5) \mathbf{A}\mathbf{C}:
This results in the matrix \begin{pmatrix} -1 & 1 \\ -3 & 3\end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 3\end{pmatrix}. The rank of this matrix is still 1.
6) \mathbf{B}\mathbf{D}:
This results in the matrix \begin{pmatrix} 1 & -1 \\ -1 & 1\end{pmatrix} \begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}. The rank of this matrix is 0.
So, the only combination that results in a rank-2 matrix is:
\mathbf{B}\mathbf{D}
Related Questions
NOTE: before you answer this question, please be aware that there aren't meant to be any full stops...
One of the reasons given for the colonization of Georgia was to serve as a "buffer" colony. This me...
One of the reasons given for the colonization of Georgia was to serve as a "buffer" colony. This me...