Math Problems Vault

Problem 3.9.1

(a)(b)(c)(d)

Some background needed for this exercise: for an $m \times n$ matrix $\mathbf{A}$ , $\mathbf{E}_{\mathbf{A}}$ denotes its (unique) reduced row echelon form. Whenever $\mathbf{B}$ can be obtained from $\mathbf{A}$ by a sequence of elementary row operations only, we write $\mathbf{A} \overset {\text{row}}{\sim } \mathbf{B}$ , which holds if and only if $\mathbf{P}\mathbf{A} = \mathbf{B}$ for some nonsingular $\mathbf{P}$ ; similarly $\mathbf{A} \overset {\text{col}}{\sim } \mathbf{B}$ (column equivalence) holds if and only if $\mathbf{A}\mathbf{Q} = \mathbf{B}$ for some nonsingular $\mathbf{Q}$ , and $\mathbf{A} \sim \mathbf{B}$ (equivalence, allowing both row and column operations) holds if and only if $\mathbf{P}\mathbf{A}\mathbf{Q} = \mathbf{B}$ for nonsingular $\mathbf{P}$ , $\mathbf{Q}$ .

Suppose that $\mathbf{A}$ is an $m \times n$ matrix.

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(a)

If $[\mathbf{A}\mid \mathbf{I}_{m}]$ is row reduced to a matrix $[\mathbf{B}\mid \mathbf{P}]$ , explain why $\mathbf{P}$ must be a nonsingular matrix such that $\mathbf{P}\mathbf{A} = \mathbf{B}$ .

(b)

If $\left[\begin{smallmatrix} \mathbf{A} \\ \mathbf{I}_{n}\end{smallmatrix}\right]$ is column reduced to $\left[\begin{smallmatrix} \mathbf{C} \\ \mathbf{Q}\end{smallmatrix}\right]$ , explain why $\mathbf{Q}$ must be a nonsingular matrix such that $\mathbf{A}\mathbf{Q} = \mathbf{C}$ .

(c)

Find a nonsingular matrix $\mathbf{P}$ such that $\mathbf{P}\mathbf{A} = \mathbf{E}_{\mathbf{A}}$ , where

\mathbf{A} = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 7 \\ 1 & 2 & 3 & 6 \end{bmatrix}.

(d)

Find nonsingular matrices $\mathbf{P}$ and $\mathbf{Q}$ such that $\mathbf{P}\mathbf{A}\mathbf{Q}$ is in rank normal form (i.e., of the form $\mathbf{N}_{r} = \left[\begin{smallmatrix} \mathbf{I}_{r} & \mathbf{0} \\ \mathbf{0} & \mathbf{0}\end{smallmatrix}\right]$ , where $r = \operatorname {rk}\left(\mathbf{A}\right)$ ).

Problem 3.9.2

(a)(b)(c)

Consider the two matrices

\mathbf{A} = \begin{bmatrix} 2 & 2 & 0 & -1 \\ 3 & -1 & 4 & 0 \\ 0 & -8 & 8 & 3 \end{bmatrix} \qquad \text{and} \qquad \mathbf{B} = \begin{bmatrix} 2 & -6 & 8 & 2 \\ 5 & 1 & 4 & -1 \\ 3 & -9 & 12 & 3 \end{bmatrix}.

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(a)

Are $\mathbf{A}$ and $\mathbf{B}$ equivalent?

(b)

Are $\mathbf{A}$ and $\mathbf{B}$ row equivalent?

(c)

Are $\mathbf{A}$ and $\mathbf{B}$ column equivalent?

Problem 3.9.3

If $\mathbf{A} \overset {\text{row}}{\sim } \mathbf{B}$ , explain why the basic columns in $\mathbf{A}$ occupy exactly the same positions as the basic columns in $\mathbf{B}$ .

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Problem 3.9.4

A product of elementary interchange matrices -- i.e., elementary matrices of Type I -- is called a permutation matrix. If $\mathbf{P}$ is a permutation matrix, explain why $\mathbf{P}^{-1} = \mathbf{P}^{T}$ .

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Problem 3.9.5

If $\mathbf{A}_{n \times n}$ is a nonsingular matrix, which (if any) of the following statements are true?

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(a)

$\mathbf{A} \sim \mathbf{A}^{-1}$ .

(b)

$\mathbf{A} \overset {\text{row}}{\sim } \mathbf{A}^{-1}$ .

(c)

$\mathbf{A} \overset {\text{col}}{\sim } \mathbf{A}^{-1}$ .

(d)

$\mathbf{A} \sim \mathbf{I}$ .

(e)

$\mathbf{A} \overset {\text{row}}{\sim } \mathbf{I}$ .

(f)

$\mathbf{A} \overset {\text{col}}{\sim } \mathbf{I}$ .

Problem 3.9.6

Which (if any) of the following statements are true?

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(a)

$\mathbf{A} \sim \mathbf{B} \implies \mathbf{A}^{T} \sim \mathbf{B}^{T}$ .

(b)

$\mathbf{A} \overset {\text{row}}{\sim } \mathbf{B} \implies \mathbf{A}^{T} \overset {\text{col}}{\sim } \mathbf{B}^{T}$ .

(c)

$\mathbf{A} \overset {\text{row}}{\sim } \mathbf{B} \implies \mathbf{A}^{T} \overset {\text{col}}{\sim } \mathbf{B}^{T}$ .

(d)

$\mathbf{A} \overset {\text{row}}{\sim } \mathbf{B} \implies \mathbf{A} \sim \mathbf{B}$ .

(e)

$\mathbf{A} \overset {\text{col}}{\sim } \mathbf{B} \implies \mathbf{A} \sim \mathbf{B}$ .

(f)

$\mathbf{A} \sim \mathbf{B} \implies \mathbf{A} \overset {\text{row}}{\sim } \mathbf{B}$ .

Problem 3.9.7

Show that every elementary matrix of Type I can be written as a product of elementary matrices of Types II and III.

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Problem 3.9.8

If $\operatorname {rk}\left(\mathbf{A}_{m \times n}\right) = r$ , show that there exist matrices $\mathbf{B}_{m \times r}$ and $\mathbf{C}_{r \times n}$ such that $\mathbf{A} = \mathbf{B}\mathbf{C}$ , where $\operatorname {rk}\left(\mathbf{B}\right) = \operatorname {rk}\left(\mathbf{C}\right) = r$ . Such a factorization is called a full-rank factorization.

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Problem 3.9.9

Prove that $\operatorname {rk}\left(\mathbf{A}_{m \times n}\right) = 1$ if and only if there are nonzero columns $\overrightarrow {u}_{m \times 1}$ and $\overrightarrow {v}_{n \times 1}$ such that

\mathbf{A} = \overrightarrow {u}\overrightarrow {v}^{T}.

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Problem 3.9.10

Prove that if $\operatorname {rk}\left(\mathbf{A}_{n \times n}\right) = 1$ , then $\mathbf{A}^{2} = \tau \mathbf{A}$ , where $\tau = \trace (\mathbf{A})$ .

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