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4.5. Exercises

Exercise 4.5.1.

Calculate the following determinant in two ways: first expand it about the first row, then expand it about the third column.

\begin{vmatrix}7&-5&3\\ -1&2&0\\ 4&4&0\end{vmatrix}

Exercise 4.5.2.

Evaluate the following determinants. You may use computer programs (such as R) to check your answer. You will be expected to show your work. These are the same matrices as Exercise 3.4.1.

(i)

$\begin{vmatrix}1&2&3\\ 2&3&4\\ 3&4&6\end{vmatrix}$
(ii)

$\begin{vmatrix}1&3&1\\ 2&6&3\\ -1&2&5\end{vmatrix}$
(iii)

$\begin{vmatrix}1&2&0\\ 0&0&1\\ 1&-2&3\end{vmatrix}$
(iv)

$\begin{vmatrix}1&0&-1\\ 2&-2&0\\ 0&3&-3\end{vmatrix}$
(v)

$\begin{vmatrix}0&2&-3\\ 1&1&1\\ -1&2&0\end{vmatrix}$
(vi)

$\begin{vmatrix}1&2&4&0\\ 0&1&1&3\\ 2&1&0&1\\ 0&1&4&-2\end{vmatrix}$
(vii)

$\begin{vmatrix}1&0&-1&0\\ 2&1&-2&0\\ 0&0&3&-3\\ 1&2&1&2\end{vmatrix}$
(viii)

$\begin{vmatrix}3&-1&0&2&1\\ 2&-2&-2&3&0\\ 1&0&1&-1&0\\ 2&1&3&-2&1\\ 0&1&0&0&0\end{vmatrix}$
(ix)

$\begin{vmatrix}3&2&1&0&1\\ 0&1&2&0&1\\ 1&1&1&1&1\\ 2&4&3&4&2\\ 1&0&2&1&2\end{vmatrix}$

Exercise 4.5.3.

For each matrix $A$ in Exercise 4.5.2 which is invertible, use Exercise 3.4.2 to write $A=L^{\prime}_{1}\cdots L^{\prime}_{k}$ as a product of elementary matrices. Then calculate the determinant of $A$ by finding $\det L^{\prime}_{1}\cdots\det L^{\prime}_{k}$ .

Exercise 4.5.4.

In each of the following questions, find all the real numbers $x$ such that $A$ is invertible without using the algorithm for calculating ${A}^{-1}$ .

(i)

$A=\begin{pmatrix}x&1\\ 0&x\end{pmatrix}$
(ii)

$A=\begin{pmatrix}2&x-1\\ -1&1\end{pmatrix}$
(iii)

$A=\begin{pmatrix}1&2x-1\\ 3&x^{2}\end{pmatrix}$
(iv)

$A=\begin{pmatrix}x-5&-4\\ 2x-1&x^{2}\end{pmatrix}$
(v)

$A=\begin{pmatrix}1-x&2x\\ x&1+x\end{pmatrix}$
(vi)

$A=\begin{pmatrix}x&1&-1\\ 2&0&1\\ -1&1&3\end{pmatrix}$
(vii)

$A=\begin{pmatrix}3&1&x\\ 1&-3&0\\ 0&x&1\end{pmatrix}$
(viii)

$A=\begin{pmatrix}1&x&-2\\ -4&x+1&0\\ -1&0&1\end{pmatrix}$

Exercise 4.5.5.

Let $A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\operatorname{M}_{{2}}({{\mathbb{R}}})$ . Prove that $A$ is not invertible if and only if there exists $\lambda\in{\mathbb{R}}$ such that

\begin{pmatrix}c\\ d\end{pmatrix}=\lambda\begin{pmatrix}a\\ b\end{pmatrix}\quad\hbox{ or }\quad\lambda\begin{pmatrix}c\\ d\end{pmatrix}=\begin{pmatrix}a\\ b\end{pmatrix}

which holds if and only if there exists $\mu\in{\mathbb{R}}$ such that

\begin{pmatrix}b\\ d\end{pmatrix}=\mu\begin{pmatrix}a\\ c\end{pmatrix}\quad\hbox{ or }\quad\mu\begin{pmatrix}b\\ d\end{pmatrix}=\begin{pmatrix}a\\ c\end{pmatrix}.

Exercise 4.5.6.

Without using the results of this section, other than the formula for the $2\times 2$ determinant, prove that for matrices $A,B\in\operatorname{M}_{{2}}({{\mathbb{R}}})$ we have

\det(AB)=\det A\det B.

Exercise 4.5.7.

For which values of $t\in{\mathbb{R}}$ is the following matrix rank 2? $\displaystyle\begin{pmatrix}1&t&0\\ 2&1&1\\ 0&3&-1\end{pmatrix}$

Exercise 4.5.8 (Implicit function theorem).

Recall the definition of a Jacobian matrix in Exercise 1.6.19. Let $F:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be a smooth multivariable function; the implicit function theorem states that if the Jacobian matrix $J_{F}$ at a point $P\in{\mathbb{R}}^{n}$ is invertible, then one can write down a “local inverse” of $F$ .

Determine the set of all points $P\in{\mathbb{R}}^{n}$ at which the Jacobian matrix is invertible, where $F$ is the following:

(i)

$F(x)=\sin x$
(ii)

$F(x,y)=(x^{2},y^{3})$
(iii)

$F(x,y)=(xy,x+y)$
(iv)

$F(x,y,z)=(xy,x^{2}z^{2},x^{2}y+z).$

Exercise 4.5.9 (Hessian matrix).

Given a smooth multivariable function $F:{\mathbb{R}}^{n}\to{\mathbb{R}}$ , the Hessian matrix associated to it is the matrix $H=(\partial^{2}F/\partial x_{i}\partial x_{j})=(F_{x_{i}x_{j}})$ of second-order partial derivatives of $F$ .

(i)

Find the Hessian matrix of $F(x,y)=xy^{2}+x^{2}$ .
(ii)

Is the Hessian matrix symmetric for any $F$ ?
(iii)

What other name is given to the quantity $\det H=F_{xx}F_{yy}-F_{xy}^{2}$ ?

Exercise 4.5.10 (Cramer’s rule).

In 1750, Swiss mathematician Gabriel Cramer published a method which can be used for inverting matrices by using determinants. Let $A$ by an $n\times n$ matrix, such that $\det A\neq 0$ . Let $C:=(A_{ij})$ be the matrix of cofactors (see Definition 4.2.5). Then Cramer’s Rule says:

A^{-1}=\frac{1}{\det A}C^{t}.

In other words, the inverse of $A$ is the transpose of the matrix of cofactors divided by the determinant of $A$ .

(i)

Verify Cramer’s rule is true by recalculating two of the inverses you found in the exercises from Chapter 3.
(ii)

Verify Cramer’s rule is true when $A$ is an elementary matrix.