Determinants

Use the definition to evaluate $\operatorname {det}\left(\mathbf{A}\right)$ for each of the following matrices.

What is the volume of the parallelepiped generated by the three vectors $\overrightarrow {x}_{1} = (3,0,-4,0)^{T}$, $\overrightarrow {x}_{2} = (0,2,0,-2)^{T}$, and $\overrightarrow {x}_{3} = (0,1,0,1)^{T}$?

Using Gaussian elimination to reduce $\mathbf{A}$ to an upper-triangular matrix, evaluate $\operatorname {det}\left(\mathbf{A}\right)$ for each of the following matrices.

Use determinants to compute the rank of $\mathbf{A} = \begin{pmatrix} 1 & 3 & -2 \\ 0 & 1 & 2 \\ -1 & -1 & 6 \\ 2 & 5 & -6 \end{pmatrix}$.

Use determinants to find the values of $\alpha$ for which the following system possesses a unique solution.

If $\mathbf{A}$ is nonsingular, explain why $\operatorname {det}\left(\mathbf{A}^{-1}\right) = 1/\operatorname {det}\left(\mathbf{A}\right)$.

Explain why determinants are invariant under similarity transformations. That is, show $\operatorname {det}\left(\mathbf{P}^{-1}\mathbf{A}\mathbf{P}\right) = \operatorname {det}\left(\mathbf{A}\right)$ for all nonsingular $\mathbf{P}$.

Explain why $\operatorname {det}\left(\mathbf{A}^{*}\right) = \overline{\operatorname {det}\left(\mathbf{A}\right)}$.

Prove that if $\mathbf{A}$ is $m \times n$, then $\operatorname {det}\left(\mathbf{A}^{*}\mathbf{A}\right) \geq 0$, and explain why $\operatorname {det}\left(\mathbf{A}^{*}\mathbf{A}\right) > 0$ if and only if $\operatorname {rk}\left(\mathbf{A}\right) = n$.

If $\mathbf{A}$ is $n \times n$, explain why $\operatorname {det}\left(\alpha \mathbf{A}\right) = \alpha^{n}\operatorname {det}\left(\mathbf{A}\right)$ for all scalars $\alpha$.

If $\mathbf{A}$ is an $n \times n$ skew-symmetric matrix, prove that $\mathbf{A}$ is singular whenever $n$ is odd.

How can you build random integer matrices with $\operatorname {det}\left(\mathbf{A}\right) = 1$?

If the $k^{th}$ row of $\mathbf{A}_{n \times n}$ is written as a sum $\mathbf{A}_{k*} = \overrightarrow {x}^{T} + \overrightarrow {y}^{T} + \cdots + \overrightarrow {z}^{T}$, where $\overrightarrow {x}^{T}, \overrightarrow {y}^{T}, \ldots , \overrightarrow {z}^{T}$ are row vectors, explain why

The CBS inequality (Cauchy--Bunyakovskii--Schwarz inequality) says that $\left|\overrightarrow {x}^{*}\overrightarrow {y}\right| \leq \left\| \overrightarrow {x}\right\|_{2} \left\| \overrightarrow {y}\right\|_{2}$ for vectors $\overrightarrow {x}, \overrightarrow {y} \in \mathbb {C}^{n \times 1}$, with equality if and only if $\overrightarrow {y} = \alpha \overrightarrow {x}$ for $\alpha = \overrightarrow {x}^{*}\overrightarrow {y}/\overrightarrow {x}^{*}\overrightarrow {x}$. Use Exercise 6.1.10 to give an alternate proof of the CBS inequality along with an alternate explanation of why equality holds if and only if $\overrightarrow {y}$ is a scalar multiple of $\overrightarrow {x}$.

Let $\mathbf{A}_{k}$ be the $k \times k$ leading principal submatrix of $\mathbf{A}_{n \times n}$ --- i.e., the submatrix taken from the upper-left-hand corner of $\mathbf{A}$ consisting of its first $k$ rows and columns. Prove that if $\mathbf{A}$ has an LU factorization $\mathbf{A} = \mathbf{L}\mathbf{U}$, then $\operatorname {det}\left(\mathbf{A}_{k}\right) = u_{11}u_{22} \cdots u_{kk}$, and deduce that the $k^{th}$ pivot is

Prove that if $\operatorname {rk}\left(\mathbf{A}_{m \times n}\right) = n$, then $\mathbf{A}^{T}\mathbf{A}$ has an LU factorization with positive pivots --- i.e., $\mathbf{A}^{T}\mathbf{A}$ is positive definite, meaning that the symmetric matrix $\mathbf{A}^{T}\mathbf{A}$ possesses an LU factorization in which every pivot is positive.

Let $\mathbf{A}(x) = \begin{pmatrix} 2-x & 3 & 4 \\ 0 & 4-x & -5 \\ 1 & -1 & 3-x \end{pmatrix}$.

When the entries of $\mathbf{A} = [a_{ij}(x)]$ are differentiable functions of $x$, we define $d\mathbf{A}/dx = [da_{ij}/dx]$ (the matrix of derivatives). For square matrices, is it always the case that $d\bigl(\operatorname {det}\left(\mathbf{A}\right)\bigr)/dx = \operatorname {det}\left(d\mathbf{A}/dx\right)$?

For a set of functions $\mathcal{S} = \left\{ f_{1}(x), f_{2}(x), \ldots , f_{n}(x)\right\}$ that are $n-1$ times differentiable, the determinant

is called the Wronskian of $\mathcal{S}$. If $\mathcal{S}$ is a linearly dependent set, explain why $w(x) = 0$ for every value of $x$.

Consider evaluating an $n \times n$ determinant from the definition (6.1.1).

Use a cofactor expansion to evaluate each of the following determinants.

Use determinants to compute the following inverses.

Is the following equation a valid derivation of Cramer's rule for solving a nonsingular system $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$, where $\mathbf{A}_{i}$ denotes the matrix identical to $\mathbf{A}$ except that column $\mathbf{A}_{*i}$ has been replaced by $\overrightarrow {b}$?

Suppose $\operatorname {rk}\left(\mathbf{B}_{m \times n}\right) = n$, and let $\mathbf{Q}$ be the orthogonal projector onto $N(\mathbf{B}^{T})$ --- i.e., $\mathbf{Q} = \mathbf{I} - \mathbf{B}(\mathbf{B}^{T}\mathbf{B})^{-1}\mathbf{B}^{T}$, the projector that annihilates $R(\mathbf{B})$ and reproduces vectors already in $N(\mathbf{B}^{T})$. For $\mathbf{A} = [\mathbf{B} \mid \overrightarrow {c}_{n \times 1}]$, prove $\overrightarrow {c}^{T}\mathbf{Q}\overrightarrow {c} = \operatorname {det}\left(\mathbf{A}^{T}\mathbf{A}\right)/\operatorname {det}\left(\mathbf{B}^{T}\mathbf{B}\right)$.

If $\mathbf{A}_{n \times n}$ is a nonsingular matrix, and if $\mathbf{D}$ and $\mathbf{C}$ are $n \times k$ matrices, explain how to use (6.2.1) to derive the formula

Explain why $\mathbf{A}$ is singular if and only if $\mathbf{A}[\operatorname {adj}(\mathbf{A})] = \mathbf{0}$.

For a nonsingular linear system $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$, explain why each component of the solution must vary continuously with the entries of $\mathbf{A}$.

For scalars $\alpha$, explain why $\operatorname {adj}(\alpha \mathbf{A}) = \alpha^{n-1}\operatorname {adj}(\mathbf{A})$.

For an $n \times n$ matrix $\mathbf{A}$, prove that the following statements are true.

In 1812, Cauchy discovered the formula that says that if $\mathbf{A}$ is $n \times n$, then $\operatorname {det}\left(\operatorname {adj}(\mathbf{A})\right) = [\operatorname {det}\left(\mathbf{A}\right)]^{n-1}$. Establish Cauchy's formula.

For the following tridiagonal matrix, $\mathbf{A}_{n}$, let $D_{n} = \operatorname {det}\left(\mathbf{A}_{n}\right)$, and derive the formula $D_{n} = 2D_{n-1}-D_{n-2}$ to deduce that $D_{n} = n+1$.

By considering rank-one updated matrices, derive the following formulas.

A bordered matrix has the form $\mathbf{B} = \begin{pmatrix} \mathbf{A} & \overrightarrow {x} \\ \overrightarrow {y}^{T} & \alpha \end{pmatrix}$ in which $\mathbf{A}_{n \times n}$ is nonsingular, $\overrightarrow {x}$ is a column, $\overrightarrow {y}^{T}$ is a row, and $\alpha$ is a scalar. Explain why the following statements must be true.

If $\mathbf{B}$ is $m \times n$ and $\mathbf{C}$ is $n \times m$, explain why (6.2.1) guarantees that $\lambda^{m}\operatorname {det}\left(\lambda \mathbf{I}_{n}-\mathbf{C}\mathbf{B}\right) = \lambda^{n}\operatorname {det}\left(\lambda \mathbf{I}_{m}-\mathbf{B}\mathbf{C}\right)$ is true for all scalars $\lambda$.

For a square matrix $\mathbf{A}$ and column vectors $\overrightarrow {c}$ and $\overrightarrow {d}$, derive the following two extensions of the rank-one update formula $\operatorname {det}\left(\mathbf{A}+\overrightarrow {c}\overrightarrow {d}^{T}\right) = \operatorname {det}\left(\mathbf{A}\right)\bigl(1+\overrightarrow {d}^{T}\mathbf{A}^{-1}\overrightarrow {c}\bigr)$ (6.2.3).

Describe the determinant of an elementary reflector $\mathbf{R} = \mathbf{I} - 2\overrightarrow {u}\overrightarrow {u}^{*}/\overrightarrow {u}^{*}\overrightarrow {u}$ (a reflection about the hyperplane $\overrightarrow {u}^{\perp }$) and a plane rotation $\mathbf{P}_{ij}$ (a rotation in the $(i,j)$-plane, using $\cos \theta$ and $\sin \theta$ in positions $(i,i),(i,j),(j,i),(j,j)$ of an otherwise identity matrix), and then explain how to find $\operatorname {det}\left(\mathbf{A}\right)$ using Householder reduction (reducing $\mathbf{A}$ to upper-trapezoidal form by a sequence of elementary reflectors) and Givens reduction (the same reduction using a sequence of plane rotations instead).

Suppose that $\mathbf{A}$ is a nonsingular matrix whose entries are integers. Prove that the entries in $\mathbf{A}^{-1}$ are integers if and only if $\operatorname {det}\left(\mathbf{A}\right) = \pm 1$.

Let $\mathbf{A} = \mathbf{I} - 2\overrightarrow {u}\overrightarrow {v}^{T}$ be a matrix in which $\overrightarrow {u}$ and $\overrightarrow {v}$ are column vectors with integer entries.

Use induction to argue that a cofactor expansion of $\operatorname {det}\left(\mathbf{A}_{n \times n}\right)$ requires

multiplications for $n \geq 2$. Assume a computer will do 1,000,000 multiplications per second, and neglect all other operations to estimate how long it will take to evaluate the determinant of a $100 \times 100$ matrix using cofactor expansions.

Determine all values of $\lambda$ for which the matrix $\mathbf{A}-\lambda \mathbf{I}$ is singular, where

Suppose that $f_{1}(t), f_{2}(t), \ldots , f_{n}(t)$ are solutions of the $n^{th}$-order linear differential equation $y^{(n)}+p_{1}(t)y^{(n-1)}+\cdots +p_{n-1}(t)y'+p_{n}(t)y = 0$, and let $w(t)$ be the Wronskian

By converting the $n^{th}$-order equation into a system of $n$ first-order equations with the substitutions $x_{1} = y, x_{2} = y', \ldots , x_{n} = y^{(n-1)}$, show that $w(t) = w(\xi_{0})\, e^{-\int_{\xi_{0}}^{t} p_{1}(\xi )\, d\xi }$ for an arbitrary constant $\xi_{0}$.

Evaluate the Vandermonde determinant by showing

When is this nonzero (compare with Example 4.3.4)?

Suppose that each entry in $\mathbf{A}_{n \times n} = [a_{ij}(x)]$ is a differentiable function of a real variable $x$. Use formula (6.1.19) to derive the formula

Consider the entries of $\mathbf{A}$ to be independent variables, and use formula (6.1.19) to derive the formula

In 1772, the French mathematician Pierre-Simon Laplace (1749--1827) presented the following generalized version of the cofactor expansion. For an $n \times n$ matrix $\mathbf{A}$, let $\mathbf{A}(i_{1}i_{2} \cdots i_{k} \mid j_{1}j_{2} \cdots j_{k})$ denote the $k \times k$ submatrix of $\mathbf{A}$ that lies on the intersection of rows $i_{1},i_{2},\ldots ,i_{k}$ with columns $j_{1},j_{2},\ldots ,j_{k}$, and let $M(i_{1}i_{2} \cdots i_{k} \mid j_{1}j_{2} \cdots j_{k})$ denote the $n-k \times n-k$ minor determinant obtained by deleting rows $i_{1},i_{2},\ldots ,i_{k}$ and columns $j_{1},j_{2},\ldots ,j_{k}$ from $\mathbf{A}$. The generalized cofactor of $\mathbf{A}(i_{1} \cdots i_{k} \mid j_{1} \cdots j_{k})$ is defined to be the signed minor $\mathring {A}(i_{1} \cdots i_{k} \mid j_{1} \cdots j_{k}) = (-1)^{i_{1}+\cdots +i_{k}+j_{1}+\cdots +j_{k}}M(i_{1} \cdots i_{k} \mid j_{1} \cdots j_{k})$, and for each fixed set of row indices $1 \leq i_{1} < \cdots < i_{k} \leq n$,

Each such sum contains $\binom {n}{k}$ terms. Use Laplace's expansion to evaluate the determinant of

in terms of the first and third rows.