Linear Algebra in a Nutshell

Linear dependence, basis, and combinations of vectors in the plane, plus 2x2 matrix arithmetic, determinants, inverses, eigenvalues, and eigenvectors.

The ODE Project, Linear Algebra in a Nutshell

Status

FlagsLoading your progress…

38 exercises

Exercise 1

Given a column vector we define the transpose of $\mathbf x$ to be If find each of the following.

$A \mathbf x$
$A \mathbf y$
$\mathbf x^T \mathbf y$
$\mathbf y^T \mathbf x$

2.5

Exercise 2

If find each of the following.

$A + B$
$2A - 3B$
$AB$
$BA$
$A^{-1}$
$B^{-1}$

2.5

Exercise 3

If find each of the following.

$A + B$
$3A - 2B$
$AB$
$BA$

2.5

Exercise 4

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 3 & 4 \\ 2 & 1 \end{pmatrix}$

2.5

Exercise 5

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 6 & 3 \\ -4 & -1 \end{pmatrix}$

2.5

Exercise 6

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}$

2.5

Exercise 7

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} -1 & 6 \\ -2 & 6 \end{pmatrix}$

2.5

Exercise 8

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 3 & 1 \\ -2 & 0 \end{pmatrix}$

2.5

Exercise 9

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 1 & -2 \\ 1 & 3 \end{pmatrix}$

2.5

Exercise 10

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 2 & 0 \\ 0 & -3 \end{pmatrix}$

2.5

Exercise 11

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$

2.5

Exercise 12

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$

2.5

Exercise 13

Find the determinant of each of the matrices $A$ in Finding Determinants.

$\displaystyle A = \begin{pmatrix} 1 & -2 \\ -3 & 2 \end{pmatrix}$

2.5

Exercise 14

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

2.5

Exercise 15

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} 6 & 3 \\ -4 & -1 \end{pmatrix}$

2.5

Exercise 16

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} 3 & -7 \\ -2 & 5 \end{pmatrix}$

2.5

Exercise 17

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} 8 & 7 \\ 2 & 2 \end{pmatrix}$

2.5

Exercise 18

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} -3 & 4 \\ -2 & 5 \end{pmatrix}$

2.5

Exercise 19

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$

2.5

Exercise 20

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} 4 & 0 \\ 0 & -3 \end{pmatrix}$

2.5

Exercise 21

Find the inverse (if it exists) of each of the matrices $A$ in Finding Inverses. that is, find the matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ , where

$\displaystyle A = \begin{pmatrix} 1 & 2 \\ -3 & -6 \end{pmatrix}$

2.5

Exercise 22

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 3 & 4 \\ 2 & 1 \end{pmatrix}$

2.5

Exercise 23

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 6 & 3 \\ -4 & -1 \end{pmatrix}$

2.5

Exercise 24

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}$

2.5

Exercise 25

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} -1 & 6 \\ -2 & 6 \end{pmatrix}$

2.5

Exercise 26

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 3 & 1 \\ -2 & 0 \end{pmatrix}$

2.5

Exercise 27

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 1 & -2 \\ 1 & 3 \end{pmatrix}$

2.5

Exercise 28

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 2 & 0 \\ 0 & -3 \end{pmatrix}$

2.5

Exercise 29

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$

2.5

Exercise 30

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$

2.5

Exercise 31

For each of the matrices $A$ in Finding Eigenvalues and Eigenvectors:

Find the characteristic polynomial of $A$ .
Find all of the eigenvalues of $A$ .
Find an eigenvector for each eigenvalue of $A$ .

$\displaystyle A = \begin{pmatrix} 1 & -2 \\ -3 & 2 \end{pmatrix}$

2.5

Exercise 32

For what values of $a$ are the vectors $(2, a)$ and $(4,-1)$ linearly independent?

2.5

Exercise 33

We define the trace of a $2 \times 2$ matrix to be the sum of its diagonal entries. That is, the trace of is $\trace(A) = a + d$ . Show that $\trace(AB) = \trace(BA)$ for any $2 \times 2$ matrices $A$ and $B$ .

2.5

Exercise 34

Let $A$ and $B$ be two $2 \times 2$ matrices. Show that $\det(AB) = \det(A) \det(B)$ .

2.5

Exercise 35

Let $A$ be a $2 \times 2$ matrix. Show that $\det(A^{-1}) = 1$ .

2.5

Exercise 36

Define the $2 \times 2$ identity matrix to be Show that $AI = IA = A$ for any $2 \times 2$ matrix.

2.5

Exercise 37

An upper triangular matrix $A$ is a matrix of the form Show that $A$ has eigenvalues $\alpha$ and $\beta$ .

2.5

Exercise 38

Let ${\mathbf x} = (x_1, x_2)$ and ${\mathbf y} = (y_1, y_2)$ . Prove that ${\mathbf x}$ and ${\mathbf y}$ are linearly independent if and only if

2.5