Math Problems Vault

Problem 3.7.1

When possible, find the inverse of each of the following matrices. Check your answer by using matrix multiplication.

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(a)

$\begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix}$ .

(b)

$\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$ .

(c)

$\begin{bmatrix} 4 & -8 & 5 \\ 4 & -7 & 4 \\ 3 & -4 & 2 \end{bmatrix}$ .

(d)

$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ .

(e)

$\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3 \\ 1 & 2 & 3 & 4 \end{bmatrix}$ .

Problem 3.7.2

Find the matrix $\mathbf{X}$ such that $\mathbf{X} = \mathbf{A}\mathbf{X}+\mathbf{B}$ , where

\mathbf{A} = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{bmatrix} \qquad \text{and} \qquad \mathbf{B} = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ 3 & 3 \end{bmatrix}.

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Problem 3.7.3

(a)(b)(c)

For a square matrix $\mathbf{A}$ , explain why each of the following statements must be true.

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(a)

If $\mathbf{A}$ contains a zero row or a zero column, then $\mathbf{A}$ is singular.

(b)

If $\mathbf{A}$ contains two identical rows or two identical columns, then $\mathbf{A}$ is singular.

(c)

If one row (or column) is a multiple of another row (or column), then $\mathbf{A}$ must be singular.

Problem 3.7.4

(a)(b)

Answer each of the following questions.

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(a)

Under what conditions is a diagonal matrix nonsingular? Describe the structure of the inverse of a diagonal matrix.

(b)

Under what conditions is a triangular matrix nonsingular? Describe the structure of the inverse of a triangular matrix.

Problem 3.7.5

If $\mathbf{A}$ is nonsingular and symmetric, prove that $\mathbf{A}^{-1}$ is symmetric.

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Problem 3.7.6

If $\mathbf{A}$ is a square matrix such that $\mathbf{I}-\mathbf{A}$ is nonsingular, prove that

\mathbf{A}(\mathbf{I}-\mathbf{A})^{-1} = (\mathbf{I}-\mathbf{A})^{-1}\mathbf{A}.

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Problem 3.7.7

Prove that if $\mathbf{A}$ is $m \times n$ and $\mathbf{B}$ is $n \times m$ such that $\mathbf{A}\mathbf{B} = \mathbf{I}_{m}$ and $\mathbf{B}\mathbf{A} = \mathbf{I}_{n}$ , then $m = n$ .

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Problem 3.7.8

If $\mathbf{A}$ , $\mathbf{B}$ , and $\mathbf{A}+\mathbf{B}$ are each nonsingular, prove that

\mathbf{A}(\mathbf{A}+\mathbf{B})^{-1}\mathbf{B} = \mathbf{B}(\mathbf{A}+\mathbf{B})^{-1}\mathbf{A} = (\mathbf{A}^{-1}+\mathbf{B}^{-1})^{-1}.

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Problem 3.7.9

(a)(b)

Let $\mathbf{S}$ be a skew-symmetric matrix with real entries.

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(a)

Prove that $\mathbf{I}-\mathbf{S}$ is nonsingular.

(b)

If $\mathbf{A} = (\mathbf{I}+\mathbf{S})(\mathbf{I}-\mathbf{S})^{-1}$ , show that $\mathbf{A}^{-1} = \mathbf{A}^{T}$ .

Remark.

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For part (a): $\overrightarrow {x}^{T}\overrightarrow {x} = 0 \implies \overrightarrow {x} = \overrightarrow {0}$ .

Problem 3.7.10

(a)(b)

For matrices $\mathbf{A}_{r \times r}$ , $\mathbf{B}_{s \times s}$ , and $\mathbf{C}_{r \times s}$ such that $\mathbf{A}$ and $\mathbf{B}$ are nonsingular, verify that each of the following is true.

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(a)

$\begin{pmatrix} \mathbf{A} & \mathbf{0} \\ \mathbf{0} & \mathbf{B} \end{pmatrix}^{-1} = \begin{pmatrix} \mathbf{A}^{-1} & \mathbf{0} \\ \mathbf{0} & \mathbf{B}^{-1} \end{pmatrix}$ .

(b)

$\begin{pmatrix} \mathbf{A} & \mathbf{C} \\ \mathbf{0} & \mathbf{B} \end{pmatrix}^{-1} = \begin{pmatrix} \mathbf{A}^{-1} & -\mathbf{A}^{-1}\mathbf{C}\mathbf{B}^{-1} \\ \mathbf{0} & \mathbf{B}^{-1} \end{pmatrix}$ .

Problem 3.7.11

(a)(b)

Consider the block matrix $\begin{pmatrix} \mathbf{A}_{r \times r} & \mathbf{C}_{r \times s} \\ \mathbf{R}_{s \times r} & \mathbf{B}_{s \times s} \end{pmatrix}$ . When the indicated inverses exist, the matrices defined by

\mathbf{S} = \mathbf{B}-\mathbf{R}\mathbf{A}^{-1}\mathbf{C} \qquad \text{and} \qquad \mathbf{T} = \mathbf{A}-\mathbf{C}\mathbf{B}^{-1}\mathbf{R}

are called the Schur complements of $\mathbf{A}$ and $\mathbf{B}$ , respectively.

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(a)

If $\mathbf{A}$ and $\mathbf{S}$ are both nonsingular, verify that

\begin{pmatrix} \mathbf{A} & \mathbf{C} \\ \mathbf{R} & \mathbf{B} \end{pmatrix}^{-1} = \begin{pmatrix} \mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{C}\mathbf{S}^{-1}\mathbf{R}\mathbf{A}^{-1} & -\mathbf{A}^{-1}\mathbf{C}\mathbf{S}^{-1} \\ -\mathbf{S}^{-1}\mathbf{R}\mathbf{A}^{-1} & \mathbf{S}^{-1} \end{pmatrix}.

(b)

If $\mathbf{B}$ and $\mathbf{T}$ are nonsingular, verify that

\begin{pmatrix} \mathbf{A} & \mathbf{C} \\ \mathbf{R} & \mathbf{B} \end{pmatrix}^{-1} = \begin{pmatrix} \mathbf{T}^{-1} & -\mathbf{T}^{-1}\mathbf{C}\mathbf{B}^{-1} \\ -\mathbf{B}^{-1}\mathbf{R}\mathbf{T}^{-1} & \mathbf{B}^{-1}+\mathbf{B}^{-1}\mathbf{R}\mathbf{T}^{-1}\mathbf{C}\mathbf{B}^{-1} \end{pmatrix}.

Problem 3.7.12

Suppose that $\mathbf{A}$ , $\mathbf{B}$ , $\mathbf{C}$ , and $\mathbf{D}$ are $n \times n$ matrices such that $\mathbf{A}\mathbf{B}^{T}$ and $\mathbf{C}\mathbf{D}^{T}$ are each symmetric and $\mathbf{A}\mathbf{D}^{T}-\mathbf{B}\mathbf{C}^{T} = \mathbf{I}$ . Prove that

\mathbf{A}^{T}\mathbf{D}-\mathbf{C}^{T}\mathbf{B} = \mathbf{I}.

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