12.E: Exercises

Last updated
Save as PDF

Page ID: 299

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Calculational Exercises

1. In each of the following, ﬁnd matrices \(A, x,\) and \(b\) such that the given system of linear equations can be expressed as the single matrix equation \(Ax = b.\)

\[ (a)~~ \left. \begin{array}{ccccccc} 2x_1 &-& 3x_2& + &5x_3 &= &7 \\ 9x_1& - &x_2& +& x_3& =& -1 \\ x_1& + &5x_2& +& 4x_3 &= &0 \end{array} \right\} ~~~ (b)~~ \left. \begin{array}{ccccccccc} 4x_1&&& -& 3x_3& +& x_4& =& 1 \\ 5x_1& +& x_2&&& -& 8x_4& =& 3 \\ 2x_1& - &5x_2& + &9x_3& -& x_4& =& 0 \\ &&3x_2& - &x_3& +& 7x_4& =& 2\end{array} \right\} \]

2. In each of the following, express the matrix equation as a system of linear equations.

\[ (a) \left[ \begin{array}{ccc} 3 & -1 & 2 \\ 4 & 3 & 7 \\ -2& 1 & 5 \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right] = \left[ \begin{array}{c} 2 \\ -1 \\ 4 \end{array} \right] ~~~ (b)\left[ \begin{array}{cccc} 3 & -2 & 0&1 \\ 5 & 0 & 2 & -2\\ 3& 1 & 4&7\\ -2&5&1&6 \end{array} \right] \left[ \begin{array}{c} w \\ x\\ y\\ z \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0\\ 0\\ 0 \end{array} \right] \]

3. Suppose that \(A, B, C, D,\) and \(E\) are matrices over \(\mathbb{F}\) having the following sizes:

\[A {\it{~is~}} 4 \times 5,~~ B {\it{~is~}} 4 \times 5,~~ C {\it{~is~}} 5 \times 2,~~ D {\it{~is~}} 4 \times 2,\]

Determine whether the following matrix expressions are deﬁned, and, for those that are deﬁned, determine the size of the resulting matrix.

\[(a)~ BA ~~~(b)~ AC + D ~~~(c)~ AE + B~~~ (d)~ AB + B~~~ (e)~E(A + B)~~~ (f) E(AC)\]

4. Suppose that \(A, B, C, D,\) and \(E\) are the following matrices:

\[ A=\left[ \begin{array}{cc} 3 & 0 \\ -1 & 2 \\ 1&1 \end{array} \right],~ B= \left[ \begin{array}{cc} 4 & -1 \\ 0 & 2 \end{array} \right], ~ C= \left[ \begin{array}{ccc} 1 & 4 &2 \\ 3 & 1&5 \end{array} \right],\\ D= \left[ \begin{array}{ccc} 1 & 5 &2 \\ -1 & 0 & 1 \\ 3& 2 & 4\end{array} \right], {\it{~and ~}}E= \left[\begin{array}{ccc} 6 & 1 &3 \\ -1 & 1 & 2 \\ 4& 1 & 3\end{array} \right]. \]

Determine whether the following matrix expressions are deﬁned, and, for those that are deﬁned, compute the resulting matrix.

\((a)~ D + E~~ (b)~ D - E~~ (c)~ 5A~~ (d)~ -7C~~ (e)~ 2B - C\\
(f)~ 2E - 2D~~ (g)~ -3(D + 2E)~~ (h)~A - A~~ (i)~ AB~~ (j)~ BA\\
(k)~ (3E)D~~ (l)~ (AB)C ~~(m)~ A(BC)~~ (n)~(4B)C + 2B ~~(o)~ D - 3E\\
(p)~ CA + 2E ~~(q)~ 4E - D ~~(r)~ DD\)

5. Suppose that \(A, B,\) and \(C\) are the following matrices and that \(a = 4\) and \(b = 7.\)

\[ A= \left[ \begin{array}{ccc} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array} \right],B = \left[ \begin{array}{ccc} 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 \end{array} \right], {\it{~and~}} C = \left[ \begin{array}{ccc} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array} \right]. \]

Verify computationally that
\( (a)~ A + (B + C) = (A + B) + C ~~~(b) ~(AB)C = A(BC)\\
(c)~ (a + b)C = aC + bC ~~~(d)~ a(B - C) = aB - aC\\
(e)~ a(BC) = (aB)C = B(aC) ~~~(f)A(B - C) = AB - AC\\
(g)~ (B + C)A = BA + CA ~~~(h) a(bC) = (ab)C\\
(i)~ B - C = -C + B\)

6. Suppose that \(A\) is the matrix
\[A=\left[ \begin{array}{cc} 3 & 1 \\ 2 & 1 \end{array} \right] \]
Compute \(p(A)\), where \(p(z)\) is given by
\((a)~ p(z) = z - 2 ~~~(b)~ p(z) = 2z^2 - z + 1\\
(c)~ p(z) = z^3 - 2z + 4~~~ (d)~ p(z) = z^2 - 4z + 1\)

7. Deﬁne matrices \(A, B, C, D,\) and \(E\) by

\[ A=\left[ \begin{array}{cc} 3 & 1 \\ 2 & 1 \end{array} \right],~ B= \left[ \begin{array}{cc} 4 & -1 \\ 0 & 2 \end{array} \right], ~ C= \left[ \begin{array}{ccc} 2 & -3 &5 \\ 9 & -1&1 \\ 1&5&4\end{array} \right],\\ D= \left[ \begin{array}{ccc} 1 & 5 &2 \\ -1 & 0 & 1 \\ 3& 2 & 4\end{array} \right], {\it{~and ~}}E= \left[\begin{array}{ccc} 6 & 1 &3 \\ -1 & 1 & 2 \\ 4& 1 & 3\end{array} \right]. \]

(a) Factor each matrix into a product of elementary matrices and an RREF matrix.
(b) Find, if possible, the LU-factorization of each matrix.
(c) Determine whether or not each of these matrices is invertible, and, if possible, compute the inverse.

8. Suppose that \(A, B, C, D,\) and \(E\) are the following matrices:

\( (a)~ 2A^T + C~~~ (b)~ D^T - E^T~~~ (c)~ (D - E)^T\\
(d)~ B^T + 5C^T~~~ (e) ~\frac{1}{2}C^T - \frac{1}{4}A~~~ (f)~ B B^T\\
(g) ~3E^T - 3D^T~~~ (h)~ (2E^T - 3D^T )^T~~~ (i)~ CC^T\\
(j)~ (DA)^T~~~ (k)~ (C^TB)A^T~~~ (l)~ (2D^T - E)A\\
(m)~ (BA^T - 2C)^T~~~ (n)~ B^T (CC^T - A^TA)~~~ (o)~D^TE^T - (ED)^T\\
(p)~ trace(DD^T)~~~ (q)~trace(4E^T - D)~~~ (r)~trace(C^TA^T + 2E^T ) \)

Proof-Writing Exercises

1. Let \(n \in \mathbb{Z}_+\) be a positive integer and \(a_{i,j} \in \mathbb{F}\) be scalars for \(i, j = 1, \ldots , n.\) Prove that
the following two statements are equivalent:
(a) The trivial solution \(x_1 = \cdots = x_n = 0\) is the only solution to the homogeneous system of equations
\[ \left. \begin{array}{ccc} \sum_{k=1}^{n} a_{1,k}x_k & = & 0 \\ & \vdots & \\ \sum_{k=1}^{n} a_{n,k}x_k & = & 0 \end{array} \right\}. \]

(b) For every choice of scalars \(c_1 , \ldots , c_n \in \mathbb{F},\) there is a solution to the system of equations \[ \left. \begin{array}{ccc} \sum_{k=1}^{n} a_{1,k}x_k & = & c_1 \\ & \vdots & \\ \sum_{k=1}^{n} a_{n,k}x_k & = & c_n \end{array} \right\}. \]
2. Let \(A\) and \(B\) be any matrices.
(a) Prove that if both \(AB\) and \(BA\) are deﬁned, then \(AB\) and \(BA\) are both square matrices.
(b) Prove that if \(A\) has size \(m \times n\) and \(ABA\) is deﬁned, then \(B\) has size \(n \times m.\)
3. Suppose that \(A\) is a matrix satisfying \(A^T A = A.\) Prove that \(A\) is then a symmetric matrix and that \(A = A^2 .\)
4. Suppose \(A\) is an upper triangular matrix and that \(p(z)\) is any polynomial. Prove or give a counterexample: \(p(A)\) is a upper triangular matrix.

Contributors

Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.