Objectives: Topics
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The definition and computation of a determinant.
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The determinant of triangular matrices.
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Section 3.1 Introduction to Determinants
Handout 3.1 Introduction to Determinants
Objectives: Goals
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Compute determinants of \(n\times n\) matrices using a cofactor expansion.
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Apply theorems to compute determinants of matrices that have particular structures.
Definition 3.1 Determinant.
Suppose \(A\) is \(n\times n\) with entries \(a_{ij}\text{.}\) Then the determinant of \(A\text{,}\) written \(\det(A)\) or \(\det A\) or \(\left| A\right|\text{,}\) is defined by the following recursive formula.
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If \(n=1\text{,}\) then \(A=[a_{11}]\) and \(\operatorname{det}(A)=a_{11}\text{.}\)
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If \(n>1\text{,}\) then
\begin{equation*}
\operatorname{det}(A)
=
a_{11}\operatorname{det}(A_{11})
- a_{12}\operatorname{det}(A_{12})
+ \cdots
+ (-1)^{1+n} a_{1n}\operatorname{det}(A_{1n}),
\end{equation*}
where \(A_{ij}\) is the submatrix obtained by deleting row \(i\) and column \(j\) of \(A\text{.}\)
Example 3.2.
Compute \(\operatorname{det}\!\begin{bmatrix} a \amp b\\ c \amp d \end{bmatrix}\text{.}\)
Solution.
Using Definition 3.1, we have that
\begin{equation*}
\det\left(\begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}\right) = \begin{vmatrix} a \amp b \\ c \amp d \end{vmatrix} = a\det\left(\begin{bmatrix} d \end{bmatrix}\right) - b\det\left(\begin{bmatrix} c \end{bmatrix}\right) = ad - bc\text{.}
\end{equation*}
Example 3.3.
Use the definition to compute
\begin{equation*}
\operatorname{det}
\begin{bmatrix}
1 \amp -5 \amp 2\\
2 \amp 4 \amp -1\\
0 \amp 3 \amp 0
\end{bmatrix}.
\end{equation*}
Solution.
Using Definition 3.1 and the result from Example 3.2, we have
\(\begin{eqnarray*} \begin{vmatrix} 1 \amp -5 \amp 2 \\ 2 \amp 4 \amp -1 \\ 0 \amp 3 \amp 0 \end{vmatrix} \amp = \amp 1\begin{vmatrix} 4 \amp -1 \\ 3 \amp 0 \end{vmatrix} - (-5)\begin{vmatrix}2 \amp -1 \\ 0 \amp 0 \end{vmatrix} + 2\begin{vmatrix} 2 \amp 4 \\ 0 \amp 3 \end{vmatrix} \\ \amp = \amp 1 \Big(4(0) - (-1)(3)\Big) + 5\Big(2(0) - (-1)(0)\Big)\\ \amp \ \amp \hspace{2in} + 2\Big(2(3) - 4(0)\Big) \\ \amp = \amp 1(3) + 5(0) + 2(6) \\ \amp = \amp 15. \end{eqnarray*}\)
Example 3.4.
Use the definition to compute
\begin{equation*}
\operatorname{det}
\begin{bmatrix}
0 \amp 3 \amp 0\\
2 \amp 4 \amp -1\\
1 \amp -5 \amp 2
\end{bmatrix}.
\end{equation*}
Solution.
Note that
\(\begin{eqnarray*}\begin{vmatrix} 0 \amp 3 \amp 0 \\ 2 \amp 4 \amp -1\\ 1 \amp -5 \amp 2 \end{vmatrix} \amp = \amp 0\begin{vmatrix} 4 \amp -1 \\ -5 \amp 2 \end{vmatrix} - 3\begin{vmatrix} 2 \amp -1 \\ 1 \amp 2 \end{vmatrix} + 0\begin{vmatrix} 2 \amp 4 \\ 1 \amp -5 \end{vmatrix}\\ \amp = \amp 0 -3\Big(2(2) - (-1)(1)\Big) + 0 \\ \amp = \amp -15.\end{eqnarray*}\)
Definition 3.5 Cofactor.
For an \(n\times n\) matrix \(A\text{,}\) the \((i,j)\) cofactor is
\begin{equation*}
C_{ij} = (-1)^{i+j}\operatorname{det}(A_{ij}).
\end{equation*}
The sign pattern alternates in a checkerboard:
\begin{equation*}
\begin{bmatrix}
+ \amp - \amp + \amp - \amp \cdots\\
- \amp + \amp - \amp + \amp \cdots\\
+ \amp - \amp + \amp - \amp \cdots\\
- \amp + \amp - \amp + \amp \cdots\\
\vdots \amp \vdots \amp \vdots \amp \vdots \amp \ddots
\end{bmatrix}
\end{equation*}
Theorem 3.6 Cofactor Expansion along Any Row or Column.
The determinant of \(A\) can be computed by expanding along any row or any column. For example, expanding down the \(j\)-th column:
\begin{equation*}
\operatorname{det}(A) = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}.
\end{equation*}
This gives a practical way to compute determinants more efficiently by choosing rows or columns with many zeros.
Example 3.7.
Use Theorem 3.6 to compute the determinant of \(A\) in an efficient manner.
\begin{equation*}
A = \begin{bmatrix}
5 \amp 4 \amp 3 \amp 2\\
0 \amp 1 \amp 2 \amp 0\\
0 \amp -1 \amp 1 \amp 0\\
0 \amp 1 \amp 1 \amp 3
\end{bmatrix}.
\end{equation*}
Solution.
We will calculate the determinant by expanding along the first column. Note that
\begin{equation*}
\left| A \right| = 5\begin{vmatrix} 1 \amp 2 \amp 0 \\ -1 \amp 1 \amp 0 \\ 1 \amp 1 \amp 3 \end{vmatrix} - 0 + 0 - 0\text{.}
\end{equation*}
To calculate the determinant of the remaining \(3 \times 3\) matrix, we can expand over any row or column we choose. How about expanding over Column 3 since there are several 0’s. Note that
\begin{equation*}
\left| A \right| = 5\begin{vmatrix} 1 \amp 2 \amp 0 \\ -1 \amp 1 \amp 0 \\ 1 \amp 1 \amp 3 \end{vmatrix} = 5\left(0 - 0 + 3\begin{vmatrix} 1 \amp 2 \\ -1 \amp 1 \end{vmatrix}\right) = 5(3)(3) = 45.
\end{equation*}
Theorem 3.8 Determinant of a Triangular Matrix.
If \(A\) is triangular (upper or lower), then
\begin{equation*}
\operatorname{det}(A) = a_{11}a_{22}a_{33}\cdots a_{nn},
\end{equation*}
the product of its diagonal entries.
Example 3.9.
Compute the determinant of the following triangular matrix.
\begin{equation*}
A = \begin{bmatrix}
7 \amp \pi \amp \pi^2 \amp \pi^3 \amp \pi^4 \amp \pi^5 \amp \pi^6 \\
0 \amp 6 \amp \pi \amp \pi^2 \amp \pi^3 \amp \pi^4 \amp \pi^5 \\
0 \amp 0 \amp 5 \amp \pi \amp \pi^2 \amp \pi^3 \amp \pi^4 \\
0 \amp 0 \amp 0 \amp 4 \amp \pi \amp \pi^2 \amp \pi^3 \\
0 \amp 0 \amp 0 \amp 0 \amp 3 \amp \pi \amp \pi^2 \\
0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 2 \amp \pi\\
0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 1
\end{bmatrix}.
\end{equation*}
Solution.
By Theorem 3.8, we have that \(\left|A\right| = 7(6)(5)(4)(3)(2)(1) = 7! = 5040.\)
A naive cofactor expansion for an \(N\times N\) matrix requires on the order of \(N!\) multiplications, which grows very quickly:
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A \(10\times 10\) matrix requires roughly \(10! = 3{,}628{,}800\) multiplications.
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A \(20\times 20\) matrix requires approximately \(20!\approx 2.4\times 10^{18}\) multiplications.
This doesn’t mean that determinants are not useful.
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We will briefly study more efficient methods for computing determinants.
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We will learn some application of determinants, such as solving linear systems and calculating area/volume of certain regions.
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Determinants are used in multivariable calculus (e.g., Jacobians for change of variables in integrals).
