Math Problems Vault

Problem 13.1

(a)(b)(c)

Let

\mathbf{A} = \left(\begin{smallmatrix} a_{11} & a_{12} & a_{13} & \boxed {a_{14}} \\ \boxed {a_{21}} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & \boxed {a_{33}} & a_{34} \\ a_{41} & \boxed {a_{42}} & a_{43} & a_{44} \end{smallmatrix}\right) .

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(a)

Write out all of the pairs that can be formed from the four boxed elements of $\mathbf{A}$ .

(b)

Indicate which of the pairs from Part (a) are positive and which are negative.

(c)

Use the formula

\sigma _{n}(1, i_{1}; \ldots ; n, i_{n}) = \sigma _{n}(i_{1}, 1; \ldots ; i_{n}, n) = \phi _{n}(i_{1}, \ldots , i_{n})

(in which $i_{1}, \ldots , i_{n}$ represents an arbitrary permutation of the first $n$ positive integers) to compute the number of pairs from Part (a) that are negative, and check that the result of this computation is consistent with your answer to Part (b).

Problem 13.2

Consider the $n \times n$ matrix

\mathbf{A} = \left(\begin{smallmatrix} x+\lambda & x & \ldots & x \\ x & x+\lambda & & x \\ \vdots & & \ddots & \\ x & x & & x+\lambda \end{smallmatrix}\right) .

``Recall'' that

\left|\mathbf{S}\right| = \left|\mathbf{R}\right| \tag {*}

for any $n \times n$ matrix $\mathbf{R}$ and for any matrix $\mathbf{S}$ formed from $\mathbf{R}$ by adding to any one of its rows or columns, scalar multiples of one or more other rows or columns; and use this result to show that

\left|\mathbf{A}\right| = \lambda ^{n-1}(nx+\lambda ) .

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Problem 13.3

Let $\mathbf{A}$ represent an $n \times n$ nonsingular matrix. Show that if the elements of $\mathbf{A}$ and $\mathbf{A}^{-1}$ are all integers, then $\left|\mathbf{A}\right| = \pm 1$ .

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Problem 13.4

Let $\mathbf{T}$ represent an $m \times m$ matrix, $\mathbf{U}$ an $m \times n$ matrix, $\mathbf{V}$ an $n \times m$ matrix, and $\mathbf{W}$ an $n \times n$ matrix. Show that if $\mathbf{T}$ is nonsingular, then

\begin{vmatrix} \mathbf{V} & \mathbf{W} \\ \mathbf{T} & \mathbf{U} \end{vmatrix} = \begin{vmatrix} \mathbf{U} & \mathbf{T} \\ \mathbf{W} & \mathbf{V} \end{vmatrix} = (-1)^{mn}\left|\mathbf{T}\right|\left|\mathbf{W} - \mathbf{V}\mathbf{T}^{-1}\mathbf{U}\right|.

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Problem 13.5

(a)(b)

Compute the determinant of the $n \times n$ matrix $\mathbf{A} = \{ a_{ij}\}$ in the special case where $n = 4$ and

\mathbf{A} = \left(\begin{smallmatrix} 0 & 4 & 0 & 5 \\ 1 & 0 & -1 & 2 \\ 0 & 3 & 0 & -2 \\ 0 & 0 & -6 & 0 \end{smallmatrix}\right) .

Do so in each of the following two ways:

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(a)

by finding and summing the nonzero terms in the expression

\sum (-1)^{\phi _{n}(j_{1}, \ldots , j_{n})} a_{1j_{1}} \cdots a_{nj_{n}} \quad \text{or} \quad \sum (-1)^{\phi _{n}(i_{1}, \ldots , i_{n})} a_{i_{1}1} \cdots a_{i_{n}n},

(where $j_{1}, \ldots , j_{n}$ or $i_{1}, \ldots , i_{n}$ is a permutation of the first $n$ positive integers and the summation is over all such permutations);

(b)

by repeated expansion in terms of cofactors---use the (general) formula

\left|\mathbf{A}\right| = \sum _{j=1}^{n} a_{ij}\alpha _{ij} \quad \text{or} \quad \left|\mathbf{A}\right| = \sum _{i=1}^{n} a_{ij}\alpha _{ij}

(where $i$ or $j$ , respectively, is any integer between 1 and $n$ inclusive and where $\alpha_{ij}$ is the cofactor of $a_{ij}$ ) to expand $\left|\mathbf{A}\right|$ (in the special case) in terms of the determinants of $3 \times 3$ matrices, to expand the determinants of the $3 \times 3$ matrices in terms of the determinants of $2 \times 2$ matrices, and finally to expand the determinants of the $2 \times 2$ matrices in terms of the determinants of $1 \times 1$ matrices.

Problem 13.6

Let $\mathbf{A} = \{ a_{ij}\}$ represent an $n \times n$ matrix. Verify that if $\mathbf{A}$ is symmetric, then the matrix of cofactors (of $\mathbf{A}$ ) is also symmetric.

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Problem 13.7

(a)(b)

Let $\mathbf{A}$ represent an $n \times n$ matrix.

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(a)

Show that if $\mathbf{A}$ is singular, then $\operatorname {adj}(\mathbf{A})$ is singular.

(b)

Show that $\operatorname {det}\left(\operatorname {adj}(\mathbf{A})\right) = [\operatorname {det}\left(\mathbf{A}\right)]^{n-1}$ .

Problem 13.8

For any $n \times n$ nonsingular matrix $\mathbf{A}$ ,

\mathbf{A}^{-1} = (1/\left|\mathbf{A}\right|) \operatorname {adj}(\mathbf{A}). \tag {*}

Use formula (*) to verify that, for any $2 \times 2$ nonsingular matrix $\mathbf{A} = \left(\begin{smallmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{smallmatrix}\right)$ ,

\mathbf{A}^{-1} = (1/k) \left(\begin{smallmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{smallmatrix}\right), \tag {**}

where $k = a_{11}a_{22} - a_{12}a_{21}$ .

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Problem 13.9

(a)(b)(c)

Let

\mathbf{A} = \left(\begin{smallmatrix} 2 & 0 & -1 \\ -1 & 3 & 1 \\ 0 & -4 & 5 \end{smallmatrix}\right) .

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(a)

Compute the cofactor of each element of $\mathbf{A}$ .

(b)

Compute $\left|\mathbf{A}\right|$ by expanding $\left|\mathbf{A}\right|$ in terms of the cofactors of the elements of the second row of $\mathbf{A}$ , and then check your answer by expanding $\left|\mathbf{A}\right|$ in terms of the cofactors of the elements of the second column of $\mathbf{A}$ .

(c)

Use formula (*) of Exercise 8 to compute $\mathbf{A}^{-1}$ .

Problem 13.10

(a)(b)

Let $\mathbf{A} = \{ a_{ij}\}$ represent an $n \times n$ matrix (where $n \geq 2$ ), and let $\alpha_{ij}$ represent the cofactor of $a_{ij}$ .

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(a)

Show [by, for instance, making use of the result of Part (b) of Exercise 11.3] that if $\operatorname {rk}\left(\mathbf{A}\right) = n-1$ , then there exists a scalar $c$ such that $\operatorname {adj}(\mathbf{A}) = c\mathbf{x}\mathbf{y}'$ , where $\mathbf{x} = \{ x_{j}\}$ and $\mathbf{y} = \{ y_{i}\}$ are any nonnull $n$ -dimensional column vectors such that $\mathbf{A}\mathbf{x} = \mathbf{0}$ and $\mathbf{A}'\mathbf{y} = \mathbf{0}$ . Show also that $c$ is nonzero and is expressible as $c = \alpha_{ij}/(y_{i}x_{j})$ for any $i$ and $j$ such that $y_{i} \neq 0$ and $x_{j} \neq 0$ .

(b)

Show that if $\operatorname {rk}\left(\mathbf{A}\right) \leq n-2$ , then $\operatorname {adj}(\mathbf{A}) = \mathbf{0}$ .

Problem 13.11

Let $\mathbf{A}$ represent an $n \times n$ nonsingular matrix and $\mathbf{b}$ an $n \times 1$ vector. Show that the solution to the linear system $\mathbf{A}\mathbf{x} = \mathbf{b}$ (in $\mathbf{x}$ ) is the $n \times 1$ vector whose $j$ th component is

\left|\mathbf{A}_{j}\right| / \left|\mathbf{A}\right|,

where $\mathbf{A}_{j}$ is a matrix formed from $\mathbf{A}$ by substituting $\mathbf{b}$ for the $j$ th column of $\mathbf{A}$ ( $j = 1, \ldots , n$ ).

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Remark.

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This result is called Cramer's rule, after Gabriel Cramer (1704--1752).

Problem 13.12

(a)(b)

Let $c$ represent a scalar, let $\mathbf{x}$ and $\mathbf{y}$ represent $n \times 1$ vectors, and let $\mathbf{A}$ represent an $n \times n$ matrix.

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(a)

Show that

\begin{vmatrix} \mathbf{A} & \mathbf{y} \\ \mathbf{x}' & c \end{vmatrix} = c\left|\mathbf{A}\right| - \mathbf{x}'\operatorname {adj}(\mathbf{A})\mathbf{y}. \tag {E.1}

(b)

Show that, in the special case where $\mathbf{A}$ is nonsingular, result (E.1) can be reexpressed as

\begin{vmatrix} \mathbf{A} & \mathbf{y} \\ \mathbf{x}' & c \end{vmatrix} = \left|\mathbf{A}\right|(c - \mathbf{x}'\mathbf{A}^{-1}\mathbf{y}),

in agreement with the more general result that, for any $n \times n$ nonsingular matrix $\mathbf{T}$ , $n \times m$ matrix $\mathbf{U}$ , $m \times n$ matrix $\mathbf{V}$ , and $m \times m$ matrix $\mathbf{W}$ ,

\begin{vmatrix} \mathbf{T} & \mathbf{U} \\ \mathbf{V} & \mathbf{W} \end{vmatrix} = \begin{vmatrix} \mathbf{W} & \mathbf{V} \\ \mathbf{U} & \mathbf{T} \end{vmatrix} = \left|\mathbf{T}\right|\left|\mathbf{W} - \mathbf{V}\mathbf{T}^{-1}\mathbf{U}\right|. \tag {*}

Problem 13.13

Let $\mathbf{V}_{k}$ represent the $(n-1) \times (n-1)$ submatrix of the $n \times n$ Vandermonde matrix

\mathbf{V} = \left(\begin{smallmatrix} 1 & x_{1} & x_{1}^{2} & \ldots & x_{1}^{n-1} \\ 1 & x_{2} & x_{2}^{2} & \ldots & x_{2}^{n-1} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & x_{n} & x_{n}^{2} & \ldots & x_{n}^{n-1} \end{smallmatrix}\right)

(where $x_{1}, x_{2}, \ldots , x_{n}$ are arbitrary scalars) obtained by striking out the $k$ th row and the $n$ th (last) column (of $\mathbf{V}$ ). Show that

\left|\mathbf{V}\right| = \left|\mathbf{V}_{k}\right|(-1)^{n-k} \prod _{i \neq k} (x_{k} - x_{i}).

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Problem 13.14

Show that, for $n \times n$ matrices $\mathbf{A}$ and $\mathbf{B}$ ,

\operatorname {adj}(\mathbf{A}\mathbf{B}) = \operatorname {adj}(\mathbf{B})\operatorname {adj}(\mathbf{A}).

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Remark.

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Use the Binet-Cauchy formula to establish that the $ij$ th element of $\operatorname {adj}(\mathbf{A}\mathbf{B})$ equals the $ij$ th element of $\operatorname {adj}(\mathbf{B})\operatorname {adj}(\mathbf{A})$ .