Linear Algebra in Action

Harry Dym

Chapter 3 Additional applications of Gaussian elimination - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that if $A \in \mathbb{F}^{n \times n}$ is an invertible matrix, then there exists a permutation matrix $P$ such that
$$
P A=L D U,
$$
where $L$ is lower triangular with ones on the diagonal, $U$ is upper triangular with ones on the diagonal and $D$ is a diagonal matrix.

Nick Johnson

Numerade Educator

04:38

Problem 3

Show that if $L_1 D_1 U_1=L_2 D_2 U_2$, where $L_j, D_j$ and $U_j$ are $n \times n$ matrices of the form exhibited in Exercise 3.1, then $L_1=L_2, D_1=D_2$ and $U_1=U_2$. [HINT: Consider $L_2^{-1} L_1 D_1=D_2 U_2 U_1^{-1}$.]

Victor Salazar

Numerade Educator

Problem 3

Show that there exists a $3 \times 3$ permutation matrix $P$ and a lower triangular matrix
$$
B=\left[\begin{array}{ccc}
1 & 0 & 0 \\
b_{21} & 1 & 0 \\
b_{31} & b_{32} & 1
\end{array}\right] \quad \text { such that }\left[\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{lll}
1 & 0 & 0 \\
\alpha & 1 & 0 \\
\beta & 0 & 1
\end{array}\right]=B P
$$
if and only if $\alpha=0$.

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04:13

Problem 4

Find a permutation matrix $P$ such that $P A=L U$, where $L$ is a lower triangular invertible $3 \times 3$ matrix and $U$ is an upper triangular invertible $3 \times 3$ matrix for the matrix $A=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0\end{array}\right]$.

Nick Johnson

Numerade Educator

Problem 5

Verify items (3), (4), and (7) of Lemma 3.4.

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07:06

Problem 6

Verify item (5) of Lemma 3.4 on the basis of (3) and the law of conservation of dimension.

Donald Albin

Numerade Educator

01:50

Problem 7

Verify Lemma 3.5. [HINT: The fact that $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_k\right\}$ is a basis for $\mathcal{N}_A \Longleftrightarrow\left\{C^{-1} \mathbf{u}_1, \ldots, C^{-1} \mathbf{u}_k\right\}$ is a basis for $\mathcal{N}_{A C}$ is helpful.]

Nick Johnson

Numerade Educator

Problem 8

Give an example of a pair of matrices $A \in \mathbb{F}^{p \times q}$ and $C \in$ $\mathbb{F}^{q \times q}$ such that $C$ is invertible, but $\mathcal{N}_A \neq \mathcal{N}_{A C}$.

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Problem 9

Let $A \in \mathbb{F}^{p \times q}, B \in \mathbb{F}^{p \times p}$ and let $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_k\right\}$ be a basis for $\mathcal{R}_A$. Show that if $B$ is left invertible, then $\left\{B \mathbf{u}_1, \ldots, B \mathbf{u}_k\right\}$ is a basis for $\mathcal{R}_{B A}$.

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02:47

Problem 10

Find a pair of matrices $A \in \mathbb{F}^{p \times q}$ and $B \in \mathbb{F}^{p \times p}$ such that $B$ is not left invertible and yet $\left\{B \mathbf{u}_1, \ldots, B \mathbf{u}_k\right\}$ is a basis for $\mathcal{R}_{B A}$ for every basis $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_k\right\}$ of $\mathcal{R}_A$.

Manisha Sarker

Numerade Educator

17:28

Problem 11

Use Gaussian elimination to find $\mathcal{N}_A$ and $\mathcal{R}_A$ for each of the following choices of the matrix $A$ :
$$
\left[\begin{array}{llll}
3 & 1 & 2 & 4 \\
2 & 1 & 8 & 7 \\
3 & 2 & 6 & 1
\end{array}\right],\left[\begin{array}{rrrrr}
1 & 2 & 0 & 2 & 1 \\
-1 & -2 & 1 & 1 & 0 \\
1 & 2 & -3 & -7 & -2
\end{array}\right],\left[\begin{array}{llll}
0 & 0 & 8 & 1 \\
1 & 2 & 4 & 1 \\
2 & 3 & 0 & 0
\end{array}\right] .
$$

Vishvajeetkumar Bhaskar Batule

Numerade Educator

02:34

Problem 12

Find a basis for the span of the vectors
$$
\left[\begin{array}{l}
2 \\
3 \\
1 \\
4
\end{array}\right],\left[\begin{array}{l}
1 \\
0 \\
2 \\
1
\end{array}\right],\left[\begin{array}{r}
0 \\
3 \\
-3 \\
2
\end{array}\right],\left[\begin{array}{r}
3 \\
-3 \\
9 \\
1
\end{array}\right]
$$

Cory Glover

Numerade Educator

03:59

Problem 13

Let
$$
U=\left[\begin{array}{lllll}
\mathbf{u}_1 & \mathbf{u}_2 & \mathbf{u}_3 & \mathbf{u}_4 & \mathbf{u}_5
\end{array}\right]=\left[\begin{array}{ccccc}
1 & 3 & 5 & 7 & 4 \\
0 & 0 & 2 & 1 & 6 \\
0 & 0 & 0 & 4 & 2 \\
0 & 0 & 0 & 0 & 0
\end{array}\right] .
$$

Show that the pivot columns $\mathbf{u}_1, \mathbf{u}_3$ and $\mathbf{u}_4$ form a basis for $\mathcal{R}_U$.

Victor Salazar

Numerade Educator

02:52

Problem 14

Show that in the setting of the previous exercise, the columns $\mathbf{u}_1, \mathbf{u}_3$ and $\mathbf{u}_5$ also form a basis for $\mathcal{R}_U$ and calculate the coefficients $a, b, c$, $d, e, f$ in the two representations
$$
\left[\begin{array}{l}
3 \\
1 \\
1 \\
0
\end{array}\right]=a \mathbf{u}_1+b \mathbf{u}_3+c \mathbf{u}_4=d \mathbf{u}_1+e \mathbf{u}_3+f \mathbf{u}_4
$$
of the vector given on the left.

Victor Salazar

Numerade Educator

09:45

Problem 15

Find the inverse of the matrix $\left[\begin{array}{lll}1 & 3 & 2 \\ 2 & 4 & 1 \\ 0 & 4 & 2\end{array}\right]$ by the GaussSeidel method.

Susan Hallstrom

Numerade Educator

09:45

Problem 16

Use the Gauss-Seidel method to find the inverse of the matrix
$$
\left[\begin{array}{llll}
1 & 2 & 1 & 2 \\
2 & 1 & 2 & 1 \\
1 & 3 & 3 & 1 \\
3 & 1 & 1 & 4
\end{array}\right]
$$

Susan Hallstrom

Numerade Educator

Problem 17

Complete the proof of Lemma 3.7.

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05:37

Problem 18

Show that formula (3.8) coincides with the first Schur complement formula (1.11) in the special case that $A_{11}$ is invertible.

Similar considerations lead to a generalization of the second Schur complement formula (1.12).

Jack Chen

Numerade Educator

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Problem 19

Let $A \in \mathbb{F}^{n \times n}$ be a four block matrix with entries $A_{11} \in$ $\mathbb{F}^{p \times p}, A_{12} \in \mathbb{F}^{p \times q}, A_{21} \in \mathbb{F}^{q \times p}, A_{22} \in \mathbb{F}^{q \times q}$, where $n=p+q$. Show that if the range conditions
$$
\mathcal{R}_{A_{12}^T} \subseteq \mathcal{R}_{A_{22}^T} \quad \text { and } \quad \mathcal{R}_{A_{21}} \subseteq \mathcal{R}_{A_{22}}
$$
are in force, then $A$ admits a factorization of the form
$$
\left[\begin{array}{ll}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{array}\right]=\left[\begin{array}{cc}
I_p & M \\
O & I_q
\end{array}\right]\left[\begin{array}{cc}
A_{11}-M A_{22} N & O \\
O & A_{22}
\end{array}\right]\left[\begin{array}{cc}
I_p & O \\
N & I_q
\end{array}\right]
$$

Nick Johnson

Numerade Educator

00:36

Problem 20

Find a system of 5 equations and 3 unknowns that has exactly one solution and a system of 3 equations and 5 unknowns that has no solutions.

Ankit Gupta

Numerade Educator

Problem 21

Let $n_L=n_L(A)$ and $n_R=n_R(A)$ denote the number of left and right inverses, respectively, of a matrix $A \in \mathbb{F}^{p \times q}$. Show that the combinations $\left(n_L=0, n_R=0\right),\left(n_L=0, n_R=\infty\right),\left(n_L=1, n_R=1\right)$ and $\left(n_L=\infty, n_R=0\right)$ are possible. [HINT: To warm up, consider upper echelon matrices first.]

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01:12

Problem 22

In the notation of the previous exercise, show that the combinations $\left(n_L=0, n_R=1\right),\left(n_L=1, n_R=0\right),\left(n_L=\infty, n_R=1\right)$, $\left(n_L=1, n_R=\infty\right)$ and ( $n_L=\infty, n_R=\infty$ ) are impossible.

Nicole Smina

Numerade Educator

05:32

Problem 23

Use the method of Gaussian elimination to solve each of the following systems of linear equations when possible:
$$
\begin{aligned}
& 2 x_1+x_2+x_3=0 \quad 3 x_1+5 x_2=9 \\
& x_1-x_2+x_3=-1, \quad 2 x_1+4 x_2=7 \text {, } \\
& 3 x_1+2 x_2+2 x_3=1 \quad 3 x_1+6 x_2=5 \\
& 5 x_1+4 x_2+6 x_3+2 x_4=8 \\
& 3 x_1+2 x_2+5 x_3+7 x_4=6 \text {. } \\
& 2 x_1+x_2+3 x_3+5 x_4=1 \\
&
\end{aligned}
$$

Jingyun Wang

Numerade Educator

02:59

Problem 24

Discuss the answers obtained in the preceding exercise in terms of $\mathcal{N}_A$ and $\mathcal{R}_A$.

Chai Santi

Numerade Educator

03:17

Problem 25

Let $\mathcal{U}$ and $\mathcal{V}$ be subspaces of $\mathbb{F}^k$ and $\mathbb{F}^{\ell}$, respectively. Show that
$$
\mathcal{W}=\left\{\left[\begin{array}{l}
\mathbf{u} \\
\mathbf{v}
\end{array}\right]: \mathbf{u} \in \mathcal{U} \quad \text { and } \quad \mathbf{v} \in \mathcal{V}\right\}
$$
is a subspace of $\mathbb{F}^{k+\ell}$ and that $\operatorname{dim} \mathcal{W}=\operatorname{dim} \mathcal{U}+\operatorname{dim} \mathcal{V}$.

Viswesh Uppalapati

Numerade Educator

03:17

Problem 26

Let $\mathcal{U}_j$ be subspaces of $\mathbb{F}^{k_j}$ for $j=1, \ldots, \ell$. Show that
$$
\mathcal{W}=\left\{\left[\begin{array}{c}
\mathbf{u}_1 \\
\vdots \\
\mathbf{u}_{\ell}
\end{array}\right]: \mathbf{u}_j \in \mathcal{U}_j\right\}
$$
is a subspace of $\mathbb{F}^{k_1+\cdots+k_{\ell}}$ and that $\operatorname{dim} \mathcal{W}=\operatorname{dim} \mathcal{U}_1+\cdots+\operatorname{dim} \mathcal{U}_{\ell}$, and exhibit a basis for $\mathcal{W}$.

Viswesh Uppalapati

Numerade Educator

00:49

Problem 27

Let $A \in \mathbb{C}^{p \times n}$ and $C \in \mathbb{C}^{p \times q}$. Show that there exists at most one matrix $B \in \mathbb{C}^{n \times q}$ such that $A B=C$ and $B^H \mathbf{v}=\mathbf{0}$ for every vector $\mathbf{v} \in \mathcal{N}_A$. [HINT: The second fact in (2.25) may be helpful.]

Hast Aggarwal

Numerade Educator

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Problem 28

Let $A \in \mathbb{C}^{p \times n}$ and $C \in \mathbb{C}^{p \times q}$ and let $V \in \mathbb{C}^{n \times k}$ be a matrix whose columns form a basis for $\mathcal{N}_A$. Show that if $A B=C$, then $\widetilde{B}=\left(I_n-V\left(V^H V\right)^{-1} V^H\right) B$ meets the conditions $A=\widetilde{B} C$ and $\widetilde{B}^H \mathbf{v}=\mathbf{0}$ for every vector $\mathbf{v} \in \mathcal{N}_A$ and is the only matrix in $\mathbb{C}^{n \times q}$ to do so.

Victor Salazar

Numerade Educator

01:20

Problem 29

Let $A, X, B \in \mathbb{C}^{n \times n}$ be such that $A X=X B$. Show that if $A$ is invertible, then there exists a matrix $C \in \mathbb{C}^{n \times n}$ such that $A X=X C$ and $C$ is invertible. [HINT: If $X$ is not invertible, then, without loss of generality, you may assume that $X=\left[\begin{array}{ll}X_1 & X_2\end{array}\right]$, where the columns of $X_1$ form a basis for $\mathcal{R}_X$ and hence $A X_1=X_1 K$ and $X_2=X_1 L$ for suitably chosen matrices $K$ and $L$.]

Angelo Rendina

Numerade Educator

Problem 30

Show that if $A \in \mathbb{C}^{n \times n}$ meets the condition $A \mathbf{x}=\varphi(\mathbf{x}) \mathbf{x}$ for every vector $\mathbf{x} \in \mathbb{C}^n$, where $\varphi(\mathbf{x})$ is a scalar valued function of $\mathbf{x}$, then $A=\lambda I_n$ for some constant $\lambda \in \mathbb{C}$.

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03:20

Problem 31

Let $A \in \mathbb{C}^{n \times n}$ and suppose that $A^{k-1} \neq O$, but $A^k=O$. Show that
$$
\operatorname{rank}\left[\begin{array}{cccc}
A^{k-1} & A^{k-2} & \cdots & I_n \\
O & A^{k-1} & \cdots & A \\
\vdots & & \ddots & \vdots \\
O & O & \cdots & A^{k-1}
\end{array}\right]=n .
$$

Oscar Bender-Stone

Numerade Educator

03:04

Problem 32

If $|\alpha| \neq 1$, then the matrix equation
$$
\left[\begin{array}{cccc}
1 & \bar{\alpha} & \ldots & \bar{\alpha}^k \\
\alpha & 1 & \ldots & \bar{\alpha}^{k-1} \\
\vdots & & & \vdots \\
\alpha^k & \alpha^{k-1} & \ldots & 1
\end{array}\right]\left[\begin{array}{c}
x_0 \\
x_1 \\
\vdots \\
x_k
\end{array}\right]=\left[\begin{array}{c}
0 \\
\vdots \\
0 \\
1
\end{array}\right]
$$
admits a unique solution $\mathbf{x} \in \mathbb{C}^{k+1}$ with bottom entry $x_k=\left(1-|\alpha|^2\right)^{-1}$. Use Gaussian elimination to verify this statement when $k=2$ and $k=3$.

Charles Carter

Numerade Educator

Problem 33

Show that if $A \in \mathbb{F}^{p \times q}, B \in \mathbb{F}^{q \times p}$ and $A B$ is invertible, then $A$ is right invertible and $B$ is left invertible, however, the converse is false.

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