Definition 3.1.1.
\begin{equation*}
\det\left( \begin{array}{rr} a & b \\ c & d \end{array} \right) = ad-bc.
\end{equation*}
Section 3.1 Determinant
We will
- define a function on matrices,
- discover an efficient algorithm for computing it, and
- expand the Big Theorem incorporating this function.
Subsection 3.1.1 Definition of Determinant
For this course we will define the determinant as a calculation done recursively.
- First we will define the determinant for \(2 \times 2 \) matrices.
- Next we will define a recursive process for computing determinants of larger matrices.
- For example the determinant of a \(3 \times 3\) matrix will involve computing determinants of 3 \(2 \times 2\) matrices.
- Similarly the determinant of a \(4 \times 4\) matrix will involve computing determinants of 4 \(3 \times 3\) determinants which each involve computing 3 determinants of \(2 \times 2\) matrices.
- How many \(2 \times 2\) determinants are required to compute the determinant of a \(5 \times 5\) matrix?
Example 3.1.2.
\(\det\left( \begin{array}{rr} 5 & 3 \\ -1 & 7 \end{array} \right) = 5(7)-(-1)3 = 38\text{.}\)
Checkpoint 3.1.3.
Compute the determinant of \(\left[ \begin{array}{rr} 2 & 9 \\ 8 & 4 \end{array} \right] \)
Next we define a cofactor. This is just a signed determinant of a sub-matrix. It has no significance outside calculating determinants.
Definition 3.1.4.
The cofactor of position \(i,j\) is the determinant of the matrix after removing row \(i\) and column \(j\text{.}\) In the diagram the removed positions are in gray.
\begin{equation*}
C_{i,j} = (-1)^{i+j} \det\left(
\begin{array}{rrcrcr}
a_{1,1} & a_{1,2}, & \ldots & \color{red}{a_{1,j}} & \ldots & a_{1,n} \\
a_{2,1} & a_{2,2}, & \ldots & \color{red}{a_{2,j}} & \ldots & a_{2,n} \\
\vdots \\
\color{red}{a_{i,1}} & \color{red}{a_{i,2}}, & \ldots & \color{red}{a_{i,j}} & \ldots & \color{red}{a_{i,n}} \\
\vdots \\
a_{m,1} & a_{m,2}, & \ldots & \color{red}{a_{m,j}} & \ldots & a_{m,n} \\
\end{array}
\right)
\end{equation*}
Note that the sign (\((-1)^{i+j}\)) of cofactors alternates in a checkerboard pattern. For a \(4 \times 4\) is looks like
\begin{equation*}
\begin{array}{rrrr}
+ & - & + & - \\
- & + & - & + \\
+ & - & + & - \\
- & + & - & +
\end{array}
\end{equation*}
Definition 3.1.5.
One way to compute a determinant is the following formula.
\begin{equation*}
\det\left( \begin{array}{rrcr} a_{1,1} & a_{1,2} & \ldots & a_{1,n} \\ a_{2,1} & a_{2,2} & \ldots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \ldots & a_{m,n} \end{array} \right) = a_{1,1} C_{1,1} + a_{1,2} C_{1,2} + \ldots + a_{1,n} C_{1,n}
\end{equation*}
Note that the elements of the first row are multiplied by their cofactors. One can use this formula with any row or any column.
Example 3.1.6.
Calculate the determinant.
\begin{align*}
\det\left( \begin{array}{rrr} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 8 & 4 & 3 \end{array} \right) & = 1(1 \cdot 3 - 4 \cdot 5)-2(1 \cdot 3 - 8 \cdot 5)+3(1 \cdot 4 - 8 \cdot 1)\\
& = 1(-17)-2(37)+3(-4)\\
& = -103
\end{align*}
Checkpoint 3.1.7.
Based on this definition, how much fun do you expect it to be to calculate by hand a determinant of a \(7 \times 7\) matrix?
Example 3.1.8.
Calculate the determinant.
\begin{align*}
\det\left( \begin{array}{rrr} 10 & 2 & 0 \\ 7 & 9 & -5 \\ 3 & 6 & 0 \end{array} \right) & = 0(\text{stuff})-(-5)(10(6)-3(2))+0(\text{stuff})\\
& = 5(54)\\
& = 270.
\end{align*}
Subsection 3.1.2 Efficient Algorithm
This activity will demonstrate the effect of changes to a matrix on the determinant. These effects can be used to more efficiently calculate determinants.
Activity 3.1.1.
Use the following matrices for this activity. \(A=\left[ \begin{array}{rrr} 1 & -1 & 8 \\ -7 & 1 & 9 \\ 3 & 1 & 4 \end{array} \right]\text{,}\) \(B=\left[ \begin{array}{rrr} 1 & -1 & 8 \\ -7 & 1 & 9 \\ 6 & 2 & 8 \end{array} \right]\text{,}\) \(C=\left[ \begin{array}{rrr} 1 & -1 & 8 \\ 3 & 1 & 4 \\ -7 & 1 & 9 \end{array} \right]\text{,}\) \(D=\left[ \begin{array}{rrr} 1 & -1 & 8 \\ -5 & -1 & 25 \\ 3 & 1 & 4 \end{array} \right]\text{.}\)
Note
- \(B\) is the result of the row operation \(R_3 \leftarrow 2R_3\) applied to matrix \(A\)
- \(C\) is the result of the row operation \(R_2 \leftrightarrow R_3\) applied to matrix \(A\)
- \(D\) is the result of the row operation \(R_2 \leftarrow 2R_1+R_2\) applied to matrix \(A\)
(a)
Calculate the determinant of each of the matrices.
(b)
For each matrix/row operation determine how the value of the determinant was changed.
(c)
Based on these three differences, what process that we know could be used to more efficiently compute the determinant?
Subsection 3.1.3 Additional Theorems
These theorems are not dependent on the previous algorithm. Use these experiments to notice additional properties.
Activity 3.1.2.
Use the following matrices. \(A = \left[ \begin{array}{rrr} 1 & 2 & 8 \\ 1 & 0 & 2 \\ 0 & 0 & 0 \end{array} \right]\text{,}\) \(B = \left[ \begin{array}{rr} 8 & 2 \\ 0 & 3 \end{array} \right]\text{,}\) and \(C = \left[ \begin{array}{rr} 9 & 0 \\ 0 & 2 \end{array} \right]\)
(a)
Calculate \(\det\left( A \right) \)
(b)
Using the previous calculation conjecture a statement about the determinants of non-invertible matrices.
(c)
Compute \(\det B\) and \(\det B^T\text{.}\)
(d)
Conjecture a theorem about \(\det B\) and \(\det B^T\text{.}\) Explain why this is true using the defining property of the determinant.
(e)
Calculate \(\det\left( C \right)\text{.}\)
(f)
Calculate \(\det(BC)\text{.}\)
(g)
Conjecture a general relationship between \(\det B,\) \(\det C,\) and \(\det(BC)\text{.}\)
