7.6: Review Problems
- Page ID
- 1970
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Find formulas for the inverses of the following matrices, when they are not singular:
a) \(\begin{pmatrix}
1 & a & b \\
0 & 1 & c \\
0 & 0 & 1 \\
\end{pmatrix}\)
b) \(\begin{pmatrix}
a & b & c \\
0 & d & e \\
0 & 0 & f \\
\end{pmatrix}\)
When are these matrices singular?
2. Write down all \(2\times 2\) bit matrices and decide which of them are singular. For those which are not singular, pair them with their inverse.
3. Let \(M\) be a square matrix. Explain why the following statements are equivalent:
a) \(MX=V\) has a \(\textit{unique}\) solution for every column vector \(V\).
b) \(M\) is non-singular.
Hint: In general for problems like this, think about the key words:
First, suppose that there is some column vector \(V\) such that the equation \(MX=V\) has two distinct solutions. Show that \(M\) must be singular; that is, show that \(M\) can have no inverse.
Next, suppose that there is some column vector \(V\) such that the equation \(MX=V\) has no solutions. Show that \(M\) must be singular.
Finally, suppose that \(M\) is non-singular. Show that no matter what the column vector \(V\) is, there is a unique solution to \(MX=V.\)
4. Left and Right Inverses: So far we have only talked about inverses of square matrices. This problem will explore the notion of
a left and right inverse for a matrix that is not square. Let
$$
A=\begin{pmatrix}0 & 1 & 1 \\ 1&1&0\end{pmatrix}
$$
a) Compute:
(i) \(A A^{T}\),
(ii) \(\big(A A^{T}\big)^{-1}\),
(iii) \(B:=A^{T} \big(A A^{T}\big)^{-1}\)
b) Show that the matrix \(B\) above is a \(\textit{right inverse}\) for \(A\), \(\textit{i.e.}\), verify that
$$
AB=I\, .
$$
c) Does \(BA\) make sense? (Why not?)
d) Let \(A\) be an \(n\times m\) matrix with \(n>m\). Suggest a formula for a left inverse \(C\) such that
$$
CA=I
$$
\(\textit{Hint: you may assume that $A^TA$ has an inverse.}\)
e) Test your proposal for a left inverse for the simple example
$$
A=\begin{pmatrix}1\\2\end{pmatrix}\, ,
$$
f) True or false: Left and right inverses are unique. If false give a counterexample.
5. Show that if the range (remember that the range of a function is the set of all its possible outputs) of a \(3\times3\) matrix \(M\) (viewed as a function \(\mathbb{R}^{3}\to \mathbb{R}^{3}\)) is a plane then one of the columns is a sum of multiples of the other columns. Show that this relationship is preserved under EROs. Show, further, that the solutions to \(Mx=0\) describe this relationship between the columns.
6. If \(M\) and \(N\) are square matrices of the same size such that \(M^{-1}\) exists and \(N^{-1}\) does not exist, does \((MN)^{-1}\) exist?
7. If \(M\) is a square matrix which is not invertible, is \(\exp{M}\) invertible?
8. Elementary Column Operations (ECOs) can be defined in the same 3 types as EROs. Describe the 3 kinds of ECOs. Show that if maximal elimination using ECOs is performed on a square matrix and a column of zeros is obtained then that matrix is not invertible.