Homework # 4 - Math 241
The problems on this homework must be done alone. The honor code is in effect.
First name: Last name:
1. If \(A\) and \(B\) are \(n \times n\) matrices such that neither is the zero matrix but \(AB\) is the zero matrix, is \(A\) an invertible matrix?
a. Yes
b. No
c. Insufficient information
2. Let$$A = \left[\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0\\\end{array}\right]$$ What is the matrix \(A^5\) (\(A\) multiplied by itself 5 times)?
a. The zero matrix
b. The identity matrix
c. Some other matrix
3. If \(A\) is a 2 by 2 matrix, and \(A^2\) is the identity matrix, must \(A\) be the identity matrix?
In questions 4-7, determine what each of the matrices do geometrically.
4. $$A = \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 2 & 0 \\ 0 & 0 & 1\end{array}\right]$$ .
a. Reflection across the \(x_1 x_3\) plane.
b. Rotation around the \(x_2\) axis.
c. Stretch in the \(x_2\) direction.
d. Flip over the plane \(x_1 - x_3 = 0\)
e. Shift by 2 in the positive \(x_2\) direction.
5. $$A = \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right]$$ .
6. $$A = \left[\begin{array}{ccc} 0 & 0 & 1\\ 0 & 1 & 0 \\ -1 & 0 & 0\end{array}\right]$$ .
7. $$A = \left[\begin{array}{ccc} 0 & 0 & 1\\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$$ .
For questions 8 - 10, say \(T: {\bf R}^5 \rightarrow {\bf R}^7\) is a linear function.
8. If \(T({\bf x}) = A{\bf x}\), then A is a
a. \(5 \times 7\) matrix
b. \(7 \times 5\) matrix
c. \(5 \times 5\) matrix
d. \(7 \times 7\) matrix
9. Is \(T\) one-to-one?
a. Yes.
b. No.
c. There is insufficient information.
10. Is \(T\) onto?
For questions 11 - 13, say \(T: {\bf R}^7 \rightarrow {\bf R}^5\) is a linear function.
11. If \(T({\bf x}) = A{\bf x}\), then A is a
12. Is \(T\) one-to-one?
13. Is \(T\) onto?
14. Say that \(F: {\bf R}^2 \rightarrow {\bf R}^2\) is the flip over the line \(x_2 = 0\), \(G: {\bf R}^2 \rightarrow {\bf R}^2\) is the flip over the line \(x_2 = 1\), \(H: {\bf R}^2 \rightarrow {\bf R}^2\) is the flip over the line \(x_2 = 3x_1\). Which of these functions are linear?
a. \(F\) only.
b. \(F\) and \(G\).
c. \(F\) and \(H\).
d. \(G\) and \(H\).
e. All of them.
15. Say \(T({\bf x}) = A{\bf x}\), and the columns of \(A\) are linearly independent. Is \(T\) one-to-one?
16. Say \(T({\bf x}) = A{\bf x}\), and the columns of \(A\) are linearly independent. Is \(T\) onto?