Chapter 3, Determinants and Matrices Video Solutions, Mathematical Methods for Physicists

Problem 1

Evaluate the following determinants:
(a) $\left|\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right|$,
(b) $\left|\begin{array}{lll}1 & 2 & 0 \\ 3 & 1 & 2 \\ 0 & 3 & 1\end{array}\right|$,
(c) $\frac{1}{\sqrt{2}}\left|\begin{array}{cccc}0 & \sqrt{3} & 0 & 0 \\ \sqrt{3} & 0 & 2 & 0 \\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0\end{array}\right|$.

Nick Johnson

Numerade Educator

02:06

Problem 2

Test the set of linear homogeneous equations
$$
x+3 y+3 z=0, \quad x-y+z=0, \quad 2 x+y+3 z=0
$$
to see if it possesses a nontrivial solution, and find one.

Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Vvn1Eec8Hpzl08Ivucuckdn8Igliwh6 Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Yfcjxtp7V4Zxsmgv8Xpg.Vn.Fy6Khx6

Numerade Educator

05:56

Problem 3

Given the pair of equations
$$
x+2 y=3, \quad 2 x+4 y=6,
$$
(a) Show that the determinant of the coefficients vanishes.
(b) Show that the numerator determinants (Eq. (3.18)) also vanish.
(c) Find at least two solutions.

Sam Stansfield

Numerade Educator

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Problem 4

Express the components of $\mathbf{A} \times \mathbf{B}$ as $2 \times 2$ determinants. Then show that the dot product $\mathbf{A} \cdot(\mathbf{A} \times \mathbf{B})$ yields a Laplacian expansion of a $3 \times 3$ determinant. Finally, note that two rows of the $3 \times 3$ determinant are identical and hence that $\mathbf{A} \cdot(\mathbf{A} \times \mathbf{B})=0$.

Lainey Roebuck

Numerade Educator

02:36

Problem 5

If $C_{i j}$ is the cofactor of element $a_{i j}$ (formed by striking out the $i$ th row and $j$ th column and including a sign $(-1)^{i+j}$ ), show that
(a) $\sum_i a_{i j} C_{i j}=\sum_l a_{j i} C_{j i}=|A|$, where $|A|$ is the determinant with the elements $a_{i j}$, (b) $\sum_i a_{i j} C_{i k}=\sum_i a_{j i} C_{k i}=0, j \neq k$.

James Kiss

Numerade Educator

00:42

Problem 6

A determinant with all elements of order unity may be surprisingly small. The Hilbert determinant $H_{i j}=(i+j-1)^{-1}, i, j=1,2, \ldots, n$ is notorious for its small values.
(a) Calculate the value of the Hilbert determinants of order $n$ for $n=1,2$, and 3 .
(b) If an appropriate subroutine is available, find the Hilbert determinants of order $n$ for $n=4,5$, and 6 .

Cheyenne Whinham

Numerade Educator

01:57

Problem 7

Solve the following set of linear simultaneous equations. Give the results to five decimal places.
$$
\begin{aligned}
1.0 x_1+0.9 x_2+0.8 x_3+0.4 x_4+0.1 x_5 & =1.0 \\
0.9 x_1+1.0 x_2+0.8 x_3+0.5 x_4+0.2 x_5+0.1 x_6 & =0.9 \\
0.8 x_1+0.8 x_2+1.0 x_3+0.7 x_4+0.4 x_5+0.2 x_6 & =0.8 \\
0.4 x_1+0.5 x_2+0.7 x_3+1.0 x_4+0.6 x_5+0.3 x_6 & =0.7 \\
0.1 x_1+0.2 x_2+0.4 x_3+0.6 x_4+1.0 x_5+0.5 x_6 & =0.6 \\
0.1 x_2+0.2 x_3+0.3 x_4+0.5 x_5+1.0 x_6 & =0.5 .
\end{aligned}
$$

James Kiss

Numerade Educator

04:16

Problem 8

Solve the linear equations $\mathbf{a} \cdot \mathbf{x}=c, \mathbf{a} \times \mathbf{x}+\mathbf{b}=0$ for $\mathbf{x}=\left(x_1, x_2, x_3\right)$ with constant vectors $\mathbf{a} \neq 0, \mathbf{b}$ and constant $c$.

Cory Glover

Numerade Educator

11:01

Problem 9

Solve the linear equations $\mathbf{a} \cdot \mathbf{x}=d, \mathbf{b} \cdot \mathbf{x}=e, \mathbf{c} \cdot \mathbf{x}=f$, for $\mathbf{x}=\left(x_1, x_2, x_3\right)$ with constant vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and constants $d, e, f$ such that $(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \neq 0$.

Naresh Bagrecha

Numerade Educator

11:01

Problem 10

Express in vector form the solution $\left(x_1, x_2, x_3\right)$ of $\mathbf{a} x_1+\mathbf{b} x_2+\mathbf{c} x_3+\mathbf{d}=0$ with constant vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ so that $(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} \neq 0$.

Naresh Bagrecha

Numerade Educator