Definition 3.1.1.
The determinant of a \(2 \times 2\) matrix \(A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}\) is denoted by \(\det(A)\) or \(|A|\) and is defined as
\begin{equation*}
\det(A) = ad - bc\text{.}
\end{equation*}
Section 3.1 Determinant
The determinant of a square matrix is a single number that encodes a wealth of information about the matrix and the linear transformation it represents. In this section, we will explore how to compute determinants efficiently and understand their fundamental properties.
We’ll begin with practical methods for calculating determinants of matrices of 2 by 2, and using Sagemath to compute larger matrices.
Beyond computation, we’ll explore the beautiful geometric meaning of determinants: they measure the signed volume (or area in 2D) of parallelepipeds formed by the column vectors of a matrix. This geometric perspective not only provides intuition but also connects linear algebra to geometry and calculus in profound ways.
Subsection 3.1.1 Fast way to get the determinant
Activity 3.1.1.
Find the determinant of the matrix
\begin{equation*}
A = \left[\begin{array}{rr}
2 \amp 1\\
3 \amp 0
\end{array}\right]\text{.}
\end{equation*}
A.det()AActivity 3.1.2.
Find the determinant of the matrix
\begin{equation*}
A = \left[\begin{array}{rrrr}
2 \amp 1 \amp -2 \amp -3 \\
3 \amp 0 \amp -1 \amp -2 \\
-3 \amp 4 \amp 1 \amp 2\\
1 \amp 3 \amp 3 \amp -1 \\
\end{array}\right]\text{.}
\end{equation*}
Activity 3.1.3. Exercise.
Find the determinant of the matrix
\begin{equation*}
B = \left[\begin{array}{rrr}
1 \amp 2 \amp 3\\
0 \amp 1 \amp 4\\
5 \amp 6 \amp 0
\end{array}\right]\text{.}
\end{equation*}
Subsection 3.1.2 Geometric Interpretation of Determinants
One of the most beautiful interpretations of the determinant is its geometric meaning: the determinant gives us the signed volume (or area in 2D) of the parallelepiped (or parallelogram in 2D) formed by the column vectors of the matrix.
Subsubsection 3.1.2.1 2D Case: Area of a Parallelogram
For a \(2 \times 2\) matrix \(A = \begin{pmatrix} a \amp c \\ b \amp d \end{pmatrix}\text{,}\) the absolute value \(|\det(A)| = |ad - bc|\) gives the area of the parallelogram formed by the vectors \(\mathbf{u} = \begin{pmatrix} a \\ b \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} c \\ d \end{pmatrix}\text{.}\)
Example 3.1.2. Area of a Parallelogram in 2D.
Consider the matrix \(A = \begin{pmatrix} 4 \amp 1 \\ 0 \amp 3 \end{pmatrix}\text{.}\) The column vectors are \(\mathbf{u} = \begin{pmatrix} 4 \\ 0 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}\text{.}\)
The determinant is:
\begin{equation*}
\det(A) = 4 \cdot 3 - 1 \cdot 0 = 12
\end{equation*}
So the area of the parallelogram is \(|\det(A)| = 12\) square units.
Insight 3.1.4.
Key Insight: The sign of the determinant tells us about orientation. If \(\det(A) > 0\text{,}\) the vectors \(\mathbf{u}\) and \(\mathbf{v}\) follow a counterclockwise orientation. If \(\det(A) < 0\text{,}\) they follow a clockwise orientation.
Subsubsection 3.1.2.2 3D Case: Volume of a Parallelepiped
For a \(3 \times 3\) matrix \(A = \begin{pmatrix} a_1 \amp b_1 \amp c_1 \\ a_2 \amp b_2 \amp c_2 \\ a_3 \amp b_3 \amp c_3 \end{pmatrix}\text{,}\) the absolute value \(|\det(A)|\) gives the volume of the parallelepiped formed by the three column vectors.
Example 3.1.5. Volume of a Parallelepiped in 3D.
Consider the matrix from the figure:
\begin{equation*}
A = \begin{pmatrix} 1 \amp 3 \amp 2 \\ 2 \amp -3 \amp -1 \\ 0 \amp 0 \amp 4 \end{pmatrix}
\end{equation*}
This gives us three vectors: \(\mathbf{u} = \langle 1, 2, 0 \rangle\text{,}\) \(\mathbf{v} = \langle 3, -3, 0 \rangle\text{,}\) and \(\mathbf{w} = \langle 2, -1, 4 \rangle\text{.}\)
Calculate the determinant:
We can verify this is correct by computing:
\begin{align*}
\det(A) \amp = 1 \cdot \begin{vmatrix} -3 \amp -1 \\ 0 \amp 4 \end{vmatrix} - 3 \cdot \begin{vmatrix} 2 \amp -1 \\ 0 \amp 4 \end{vmatrix} + 2 \cdot \begin{vmatrix} 2 \amp -3 \\ 0 \amp 0 \end{vmatrix}\\
\amp = 1(-12 - 0) - 3(8 - 0) + 2(0 - 0)\\
\amp = -12 - 24 + 0 = -36
\end{align*}
Therefore, the volume is \(|\det(A)| = 36\) cubic units.
The geometric interpretation extends to higher dimensions, though we cannot visualize it. For an \(n \times n\) matrix, \(|\det(A)|\) gives the \(n\)-dimensional "hypervolume" of the parallelepiped formed by the \(n\) column vectors.
Activity 3.1.4. Exploring Geometric Determinants.
Use the geometric interpretation to understand determinants better.
(a)
Find the area of the parallelogram formed by vectors \(\mathbf{u} = \langle 2, 1 \rangle\) and \(\mathbf{v} = \langle 3, 4 \rangle\text{.}\)
(b)
What happens to the area when you swap the two vectors (i.e., consider \(\mathbf{v}\) and \(\mathbf{u}\) instead)? What changes?
(c)
Find the volume of the parallelepiped formed by \(\mathbf{a} = \langle 1, 0, 0 \rangle\text{,}\) \(\mathbf{b} = \langle 2, 3, 0 \rangle\text{,}\) and \(\mathbf{c} = \langle 1, 1, 5 \rangle\text{.}\)
(d)
What does it mean geometrically when \(\det(A) = 0\text{?}\) Consider the vectors \(\langle 2, 4 \rangle\) and \(\langle 1, 2 \rangle\text{.}\) What is special about their parallelogram?
Theorem 3.1.7. Geometric Interpretation of Determinants.
Let \(A\) be an \(n \times n\) matrix with column vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\text{.}\) Then:
-
\(|\det(A)|\) equals the \(n\)-dimensional volume of the parallelepiped determined by \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\text{.}\)
-
\(\det(A) = 0\) if and only if the vectors are linearly dependent (i.e., the parallelepiped is "flat" or degenerate).
-
The sign of \(\det(A)\) indicates the orientation of the vectors.
