Determinant

The factor by which a linear map scales area or volume, cofactor expansion, how row operations change it, and its verdict on whether a matrix is invertible.

Notes on Diffy Qs: Differential Equations for Engineers, Determinant

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4 exercises

Exercise 101

Compute the determinant of the following matrices:

$\begin{bmatrix} -2 \end{bmatrix}$
$\begin{bmatrix} 2 & -2 \\ 1 & 3 \end{bmatrix}$
$\begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$
$\begin{bmatrix} 2 & 9 & -11 \\ 0 & -1 & 5 \\ 0 & 0 & 3 \end{bmatrix}$
$\begin{bmatrix} 2 & 1 & 0 \\ -2 & 7 & 3 \\ 1 & 1 & 0 \end{bmatrix}$
$\begin{bmatrix} 5 & 1 & 3 \\ 4 & 1 & 1 \\ 4 & 5 & 1 \end{bmatrix}$
$\begin{bmatrix} 3 & 2 & 5 & 7 \\ 0 & 0 & 2 & 0 \\ 0 & 4 & 5 & 0 \\ 2 & 1 & 2 & 4 \end{bmatrix}$
$\begin{bmatrix} 0 & 2 & 1 & 0 \\ 1 & 2 & -3 & 4 \\ 5 & 6 & -7 & 8 \\ 1 & 2 & 3 & -2 \end{bmatrix}$

2.5

Exercise 102

For which $x$ are the following matrices singular (not invertible):

$\begin{bmatrix} 1 & 3 \\ 1 & x \end{bmatrix}$
$\begin{bmatrix} 3 & x \\ 1 & 3 \end{bmatrix}$
$\begin{bmatrix} x & 3 \\ 3 & x \end{bmatrix}$
$\begin{bmatrix} x & 1 & 0 \\ 1 & 4 & 0 \\ 1 & 6 & 2 \end{bmatrix}$

2.5

Exercise 103

Compute without computing the inverse.

2.5

Exercise 104

Find all the $x$ that make the matrix inverse have only integer entries (no fractions). Note that there are two answers.

4.0