Section 3.2 Properties of Determinants
The determinant function has many remarkable algebraic properties that make it a powerful computational and theoretical tool. Understanding these properties not only helps us compute determinants more efficiently but also reveals deep connections between matrix operations and their geometric meanings.
Subsection 3.2.1 Basic Properties
This theorem establishes one of the most important connections in linear algebra: the determinant provides a simple test for invertibility. Beyond this fundamental relationship, determinants satisfy several algebraic properties that make them powerful computational tools.
Theorem 3.2.2. Fundamental Properties of Determinants.
Let \(A\) and \(B\) be \(n \times n\) matrices. Then:
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\(\det(I_n) = 1\text{,}\) where \(I_n\) is the \(n \times n\) identity matrix.
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\(\det(A^T) = \det(A)\text{,}\) where \(A^T\) is the transpose of \(A\text{.}\)
-
\(\det(AB) = \det(A) \cdot \det(B)\) (multiplicative property).
-
If \(A\) is invertible, then \(\det(A^{-1}) = \frac{1}{\det(A)}\text{.}\)
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For any scalar \(c\text{,}\) \(\det(cA) = c^n \det(A)\text{.}\)
Example 3.2.3. Using Multiplicative and Transpose Properties.
Given two matrices \(A\) and \(B\text{,}\) use the properties from Theorem 3.2.2 to answer the following questions.
Let \(A = \begin{pmatrix} 2 \amp 1 \amp -1 \\ 0 \amp 3 \amp 2 \\ 1 \amp -1 \amp 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 \amp 2 \amp 0 \\ -1 \amp 1 \amp 3 \\ 2 \amp 0 \amp 1 \end{pmatrix}\text{.}\)
Question 1: If \(\det(A) = 35\) and \(\det(B) = 14\text{,}\) what is \(\det(AB)\text{?}\) Use property (3) from Theorem 3.2.2.
Solution: By the multiplicative property, \(\det(AB) = \det(A) \cdot \det(B) = 35 \times 14 = 490\text{.}\)
Solution: By the transpose property, \(\det(A^T) = \det(A) = 35\text{.}\)
Question 3: If \(A\) is invertible with \(\det(A) = 35\text{,}\) what is \(\det(A^{-1})\text{?}\) Use property (4) from Theorem 3.2.2.
Solution: By the inverse property, \(\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{35}\text{.}\)
Question 4: Using the properties, what is \(\det(BA^T)\text{?}\)
Solution: By combining properties (2) and (3):
\begin{equation*}
\det(BA^T) = \det(B) \cdot \det(A^T) = \det(B) \cdot \det(A) = 14 \times 35 = 490
\end{equation*}
Example 3.2.4. Scalar Multiplication Property.
Use property (5) from Theorem 3.2.2 to answer questions about scalar multiplication of matrices.
Let \(A = \begin{pmatrix} 1 \amp 2 \amp 0 \\ -1 \amp 3 \amp 1 \\ 2 \amp -1 \amp 4 \end{pmatrix}\) and suppose \(\det(A) = 29\text{.}\)
Solution: By property (5), for an \(n \times n\) matrix, \(\det(cA) = c^n \det(A)\text{.}\) Since \(A\) is \(3 \times 3\text{:}\)
\begin{equation*}
\det(2A) = 2^3 \cdot \det(A) = 8 \times 29 = 232
\end{equation*}
Question 2: What is \(\det(3A)\text{?}\)
Solution: \(\det(3A) = 3^3 \cdot \det(A) = 27 \times 29 = 783\)
Question 3: What is \(\det\left(\frac{1}{2}A\right)\text{?}\)
Solution: \(\det\left(\frac{1}{2}A\right) = \left(\frac{1}{2}\right)^3 \cdot \det(A) = \frac{1}{8} \times 29 = \frac{29}{8}\)
Question 4: If \(B\) is a \(4 \times 4\) matrix with \(\det(B) = 5\text{,}\) what is \(\det(-2B)\text{?}\)
Solution: Since \(B\) is \(4 \times 4\text{:}\)
\begin{equation*}
\det(-2B) = (-2)^4 \cdot \det(B) = 16 \times 5 = 80
\end{equation*}
Insight 3.2.5.
Important Note: Notice that the exponent in \(c^n\) matches the size of the matrix. This is a common source of errors—don’t forget to raise the scalar to the power of the matrix dimension!
Subsection 3.2.2 Determinants of Special Matrices
Before we state the important theorem, let’s recall what these special matrices look like:
Definition 3.2.6. Triangular Matrices.
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An \(n \times n\) matrix \(A\) is upper triangular if all entries below the main diagonal are zero. The general form is:\begin{equation*} A = \begin{pmatrix} a_{11} \amp a_{12} \amp a_{13} \amp \cdots \amp a_{1n} \\ 0 \amp a_{22} \amp a_{23} \amp \cdots \amp a_{2n} \\ 0 \amp 0 \amp a_{33} \amp \cdots \amp a_{3n} \\ \vdots \amp \vdots \amp \vdots \amp \ddots \amp \vdots \\ 0 \amp 0 \amp 0 \amp \cdots \amp a_{nn} \end{pmatrix} \end{equation*}
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An \(n \times n\) matrix \(A\) is lower triangular if all entries above the main diagonal are zero. The general form is:\begin{equation*} A = \begin{pmatrix} a_{11} \amp 0 \amp 0 \amp \cdots \amp 0 \\ a_{21} \amp a_{22} \amp 0 \amp \cdots \amp 0 \\ a_{31} \amp a_{32} \amp a_{33} \amp \cdots \amp 0 \\ \vdots \amp \vdots \amp \vdots \amp \ddots \amp \vdots \\ a_{n1} \amp a_{n2} \amp a_{n3} \amp \cdots \amp a_{nn} \end{pmatrix} \end{equation*}
Definition 3.2.7. Diagonal Matrix.
An \(n \times n\) matrix \(A\) is diagonal if all entries off the main diagonal are zero. The general form is:
\begin{equation*}
A = \begin{pmatrix}
d_1 \amp 0 \amp 0 \amp \cdots \amp 0 \\
0 \amp d_2 \amp 0 \amp \cdots \amp 0 \\
0 \amp 0 \amp d_3 \amp \cdots \amp 0 \\
\vdots \amp \vdots \amp \vdots \amp \ddots \amp \vdots \\
0 \amp 0 \amp 0 \amp \cdots \amp d_n
\end{pmatrix}
\end{equation*}
We often write this as \(A = \text{diag}(d_1, d_2, \ldots, d_n)\text{.}\)
Note that a diagonal matrix is both upper triangular and lower triangular.
Example 3.2.8. Examples of Special Matrices.
Here are concrete examples of the special matrix types:
Upper Triangular Matrix:
\begin{equation*}
U = \begin{pmatrix}
2 \amp 3 \amp -1 \\
0 \amp 5 \amp 4 \\
0 \amp 0 \amp -2
\end{pmatrix}
\end{equation*}
Notice all entries below the main diagonal are zero.
Lower Triangular Matrix:
\begin{equation*}
L = \begin{pmatrix}
3 \amp 0 \amp 0 \\
-1 \amp 4 \amp 0 \\
2 \amp 5 \amp 6
\end{pmatrix}
\end{equation*}
Notice all entries above the main diagonal are zero.
Diagonal Matrix:
\begin{equation*}
D = \begin{pmatrix}
3 \amp 0 \amp 0 \\
0 \amp -2 \amp 0 \\
0 \amp 0 \amp 5
\end{pmatrix}
\end{equation*}
Notice all entries off the main diagonal are zero. This matrix is both upper and lower triangular.
Theorem 3.2.9. Determinants of Special Matrices.
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Triangular matrices: If \(A\) is upper or lower triangular, then \(\det(A)\) equals the product of the diagonal entries.
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Diagonal matrices: If \(A = \text{diag}(d_1, d_2, \ldots, d_n)\text{,}\) then \(\det(A) = d_1 \cdot d_2 \cdots d_n\text{.}\)
Activity 3.2.1. Computing Determinants of Special Matrices.
(a)
Compute the determinant of the upper triangular matrix
\begin{equation*}
U = \begin{pmatrix}
2 \amp 3 \amp -1 \\
0 \amp 5 \amp 4 \\
0 \amp 0 \amp -2
\end{pmatrix}
\end{equation*}
(b)
Compute the determinant of the lower triangular matrix
\begin{equation*}
L = \begin{pmatrix}
3 \amp 0 \amp 0 \\
-1 \amp 4 \amp 0 \\
2 \amp 5 \amp 6
\end{pmatrix}
\end{equation*}
(c)
Compute the determinant of the diagonal matrix
\begin{equation*}
D = \begin{pmatrix}
3 \amp 0 \amp 0 \\
0 \amp -2 \amp 0 \\
0 \amp 0 \amp 5
\end{pmatrix}
\end{equation*}
(d)
Verify your answers using SageMath by computing the determinants directly.
Activity 3.2.2. Exercise: Exploring Determinant Properties.
Practice using properties of determinants to solve problems efficiently.
(a)
If \(\det(A) = 5\) and \(\det(B) = -2\text{,}\) find:
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\(\displaystyle \det(AB)\)
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\(\displaystyle \det(A^{-1})\)
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\(\det(2A)\) (where \(A\) is \(3 \times 3\))
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\(\displaystyle \det(A^T B)\)
