1.
\(\A=\left(\begin{array}{rr}-2\amp 7\\-1\amp 4 \end{array} \right)\)
Section 2.12 Matrix Algebra: Homework
Exercises Exercises
Exercise Group.
Find the inverse, if it exists, for each of the of the following matrices. For those for which an inverse exists, show a check of your answer.
2.
\(\B=\left(\begin{array}{rr}6\amp 4\\9\amp 6 \end{array} \right)\)
3.
\(\C=\left(\begin{array}{rr}\frac{1}{2}\amp 2\\4\amp 24 \end{array} \right)\)
4.
Solve \(\A\x=\b\) where \(\A=\left(\begin{array}{rr}-2\amp1\\7\amp5\end{array}\right)\) and \(\b=\left(\begin{array}{r}-4\\9\end{array}\right).\)5.
Solve \(\A\x=\b\) where \(\A=\left(\begin{array}{rrrr}3\amp0\amp0\amp0\\0\amp8\amp0\amp0\\0\amp0\amp-2\amp0\\0\amp0\amp0\amp9\end{array}\right)\) and \(\b=\left(\begin{array}{r}15\\-19\\28\\41\end{array}\right).\)6.
Let the matrices \(\A=\left(\begin{array}{rr}1\amp -1\\2\amp 1\\3\amp 1 \end{array} \right),\) \(\C=\left(\begin{array}{rrrr}2 \amp -1 \amp 0 \amp -2\\ 0 \amp 3 \amp 2 \amp 1 \end{array} \right),\) and \(\D=\left(\begin{array}{rrrr}5 \amp 0 \amp -1 \amp 2\\ 4 \amp -1 \amp 0 \amp 3 \end{array} \right).\) Verify that the additive part of Theorem 2.10.6 holds for these matricies. That is, compute \(\A(\C+\D)\) and \(\A\C+\A\D\) and show they are equal.
7.
Let \(\A=\left(\begin{array}{rr}-1 \amp 2 \\ -2 \amp 3 \end{array} \right)\) and \(\B=\left(\begin{array}{rr}4 \amp 1 \\ 3 \amp 1 \end{array} \right).\) Verify that \(\A\) and \(\B\) satisfy Theorem 2.10.27. That is, show \((\A\B)^{-1} = \B^{-1}\A^{-1}.\)
8.
Let \(\A=\left(\begin{array}{rr}2\amp 1\\4\amp 2 \end{array} \right).\) Find a non-zero matrix \(\B \in \R^{2 \times 2}\) such that \(\A\B=0_{2 \times 2}\) or explain why no such matrix exists.
9.
Find a matrix \(\A \in \R^{2 \times 2}\) for which there does not exist a non-zero matrix \(\B \in \R^{2 \times 2}\) so that \(\A\B=0_{2 \times 2}.\) That is \(\A\B=0_{2 \times 2}\) if and only if \(\B=\boldsymbol{0}_{2 \times 2}.\) Explain carefully.
10.
Prove Theorem 2.10.9.
11.
Prove Theorem 2.10.12.
Exercise Group.
By setting up equations for their entries, find the sets of left- or right-identities for the indicated matrix.
12.
Left-identity for \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp5\amp4\end{array}\right).\)
13.
Left-identity for \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp4\amp2\end{array}\right).\)
14.
Left-identity for \(\A=\left(\begin{array}{rr}3\amp2\\6\amp5\\-4\amp-5\end{array}\right).\)
15.
Left-identity for \(\A=\left(\begin{array}{rr}3\amp2\\6\amp4\\-9\amp-6\end{array}\right).\)
16.
Right-identity for \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp5\amp4\end{array}\right).\)
17.
Right-identity for \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp4\amp2\end{array}\right).\)
18.
Right-identity for \(\A=\left(\begin{array}{rr}3\amp2\\6\amp5\\-4\amp-5\end{array}\right).\)
19.
Right-identity for \(\A=\left(\begin{array}{rr}3\amp2\\6\amp4\\-9\amp-6\end{array}\right).\)
20.
Prove Theorem 2.10.16.
21.
Prove Theorem 2.10.20.
22.
Prove Theorem 2.10.22.
23.
Prove that if \(\A\) has a left-inverse \(\B\) and a right-inverse \(\C\) then \(\A\) must be square and \(\B=\C.\)
24.
Prove Theorem 2.10.24.25.
Prove Theorem 2.10.25.26.
Prove Corollary 2.10.28.
27.
Prove that if \(\A\in\R^{2\times2}\) is invertible, then the mapping \(T_\A:\R^2\rightarrow\R^2\) given by
\begin{equation*}
T_\A\left(\vec{x}\right)=\A\vec{x}
\end{equation*}
is
- surjective
- injective
Exercise Group.
Compute the transpose of each of the following matrices.
28.
\(\A=\left(\begin{array}{r}0\\-4\\9 \end{array} \right)\)
29.
\(\B=\left(\begin{array}{rrr}2\amp 1\amp 7 \end{array} \right)\)
30.
\(\C=\left(\begin{array}{rr}3\amp 2\\6\amp 2\\3\amp 9 \end{array} \right)\)
Exercise Group.
31.
\((\B\C)^T\)
32.
\(\C^T\B^T\)
Exercise Group.
\(\A\in\R^{n\times n}\) is a normal matrix if \(\A\) commutes with its transpose \(\A^T.\)
33.
Prove that if \(\A\) is symmetric then \(\A\) is normal.
34.
Prove that if \(\A\) is skew-symmetric then \(\A\) is normal.
35.
If \(\A\) is normal, is \(\A\) necessarily symmetric? Prove it true, or provide a counterexample.
36.
Give an example of \(\A\in\R^{2\times 2}\) for which \(\A^2=\begin{pmatrix}{0}\amp 0\\0\amp 0 \end{pmatrix}\) but \(\A^T\A\ne\begin{pmatrix}{0}\amp 0\\0\amp 0 \end{pmatrix} .\)
37.
Let \(\v=\begin{pmatrix}x\\y\end{pmatrix}.\) Compute \(\v^T\left(\begin{array}{rr} -3\amp -1\\2\amp 8 \end{array} \right)\v.\)
38.
Let \(\v=\begin{pmatrix}x\\y\end{pmatrix}.\) Find two different matrices \(\A,\B\in\R^{2\times 2}\) for which
\begin{equation*}
\v^T\A\v=\v^T\B\v=x^2+3xy-6y^2.
\end{equation*}
Exercise Group.
By setting up equations for their entries, find the set of all inverses of the indicated type or explain why no such inverse exists.
39.
Left-inverse of \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp5\amp4\end{array}\right).\)
40.
Left-inverse of \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp4\amp2\end{array}\right).\)
41.
Left-inverse of \(\A=\left(\begin{array}{rr}3\amp2\\6\amp5\\-4\amp-5\end{array}\right).\)
42.
Left-inverse of \(\A=\left(\begin{array}{rr}3\amp2\\6\amp4\\-9\amp-6\end{array}\right).\)
43.
Right-inverse of \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp5\amp4\end{array}\right).\)
44.
Right-inverse of \(\A=\left(\begin{array}{rrr}3\amp2\amp1\\6\amp4\amp2\end{array}\right).\)
45.
Right-inverse of \(\A=\left(\begin{array}{rr}3\amp2\\6\amp5\\-4\amp-5\end{array}\right).\)
46.
Right-inverse of \(\A=\left(\begin{array}{rr}3\amp2\\6\amp4\\-9\amp-6\end{array}\right).\)
