3.1 Introduction to Determinants

Theorem 1

The determinant of an $n \times n$ matrix $A$ is given by the formula:

Example: Compute $\det{A}$

Theorem 2

If $A$ is a triangular matrix, then $\det{A}$ is the product of the entries on the main diagonal of $A$

Questions

4, 10, 22, 28, 34, 42

3.2 Properties of Determinants

Theorem 3

The determinant of a square matrix $A$ is affected by elementary row operations per the following rules:

Theorem 4

A square matrix $A$ is inveritble if and only if $\det{A}\neq 0$

Theorem 5

If $A$ is an $n \times n$ matrix, then $\det{A^\top} = \det{A}$

Theorem 6

If $A$ and $B$ are $n\times n$ matrices, then $\det{AB} = (\det{A})(\det{B})$

Questions

Find $\det{\begin{bmatrix} 1 & 3 & 2 & -4\\ 0 & 1 & 2 & -5\\ 2 & 7 & 6 & -3\\ -3 & -10 & -7 & 2\\ \end{bmatrix}}$

First, we can row-reduce via the following operations:

The determinant of a triangular matrix is just the product of the diagonal entries. Notice that we swapped two rows, thus we multiply the product by $(-1)$.

True or False: If $\det{A}$ is zero, then two rows or two columns are the same, or a row or a column is zero.

False. By Invertible Matrix Theorem, $\det{A}=0$ implies linear dependence. However, this isn't the correct definition.

True or False: $\det{A^{-1}} = (-1)\det{A}$

False. Most obvious counter-example is the identity matrix $I$, which serves as its own inverse

3.3 Cramer's Rule, Volume, and Linear Transformations

Theorem 7: Cramer’s Rule

Let $A$ be an invertible $n\times n$ matrix. For any $b$ in $\R^n$, the unique solution $x$ of $Ax=b$ has entries given by

where $A_i(b)$ is $A$ with the $i$-th column replaced with $b$.

Example: Use Cramer’s rule to solve the system

Theorem 8: An Inverse Formula

Let $A$ be an invertible $n\times n$ matrix. Then

where

and

Example: Find the inverse of the matrix $A=\begin{bmatrix} 2 & 1 & 3\\ 1 & -1 & 1\\ 1 & 4 & -2 \end{bmatrix}$

Theorem 9

If $A$ is a $2\times 2$ matrix, the area of the parallelogram determined by the columns of $A$ is $|\det{A}|$. This extends to $3$ dimensional volume of parallelepipeds. Note that one of the vectors must be at the origin.

Example: Calculate the area of the parallelogram determined by the points $(- 2, -2), (0, 3), (4, -1), (6, 4)$

First, we must transpose the shape such that one of the points is at the origin. So we transpose the whole shape by $(0,-3)$.

Then, choose any $2$ points non-origin vectors that are not on the same $y$-coordinate to be the defining vectors and compute the determinant.

Theorem 10

Questions

Compute the solution of the system using Cramer’s rule:

The solution $x$ to this system $Ax=b$ is a $3$-dimensional vector where each entry is computed using Cramer’s rule:

Determine the values of the parameter s for which the system has a unique solution, describe the solution.

By Invertible Matrix Theorem and Theorem 4, we know that a matrix is invertible if and only if $\det{A} \neq 0$. We know the determinant is given by $\det{A} = 15s^2-60$, which has the roots $\pm2$. Using Cramer’s rule, we can express the general solution to the system:

Compute the adjugate matrix $\text{adj}{A}$

Recall that the adjugate matrix is given by

where

So we compute each entry individually:

And we create the adjugate matrix in transpose order:

Find the area of the parallelogram defined by the following vertices: $(0,-2) (5, -2), (-3, 1), (2, 1)$

By Theorem 9, we know that the area of this parallelogram is given by the determinant of any two non-origin vectors. First, we transpose the shape to the origin, translating by $(0,2)$.

Then select any $2$ non-origin vectors that are not on the same $y$-coordinate and compute their determinant:

Note that the following two vectors do not work because they share the same $y$-coordinate:

Let $T: \R^m \rightarrow \R^n$ be a linear transformation, and let $p$ be a vector and $S$ a set in $\R^m$. Show that the image of $p+S$ under $T$ is the translated set $T(p)+T(S)$ in $\R^n$.

By definition, $p+S$ is the set of all vectors of the form $p+v$. We know that linear transformations are closed under vector addition

Thus, the above condition holds for each $v\in S$.

Compute the area of the image of $S$ under the mapping $x \mapsto Ax$

By Theorem 10 we know that the change in volume after the transformation by $A$ is equal to the determinant of $A$.