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).