Determinant Example
Gauss Example 1
What's the determinant of A?
\[ A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \]
\[ \begin{split}\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \Leftrightarrow^{L1 \leftrightarrow L2} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \Leftrightarrow^{L3 \leftrightarrow L4} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\end{split} \]The determinant is $det(A) = 1*1*1*1 = 1$ but since we swapped lines 2 times, then $det(A) = 1 * (-1)^2 = 1$.
Gauss Example 2
What's the determinant of A?
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix} \]
\[ \begin{split}\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix} \Leftrightarrow^{L_2 \leftarrow L_2 + -2*L_1 } \begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & -2 \\ 3 & 4 & 5 \end{pmatrix} \Leftrightarrow^{L_3 \leftarrow L_3 + -3*L_1 } \\ \begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & -2 \\ 0 & -2 & -4 \end{pmatrix} \Leftrightarrow^{L_3 \leftarrow L_3 + 2*L_2 } \begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & -2 \\ 0 & 0 & 0 \end{pmatrix}\end{split} \]The determinant is $det(A) = 1*-1*0 = 0$.
Cofactors Example
\[
M=\begin{pmatrix}-1 & 4 & -7 \\2 & -5 & 8 \\-3 & 6 & -9\end{pmatrix}
\]
We are picking the column $p=3 \to (-7,8,-9)$. The formula is
\[
\begin{split}
det(M)
= a_{1,3} * C_{1,3} + a_{2,3} * C_{2,3} + a_{3,3} * C_{3,3} \\
= -7 * C_{1,3} + 8 * C_{2,3} + -9 * C_{3,3}
\end{split}
\]
- $C_{1,3} = (-1)^{4} * det(M_{1,3}) = det(M_{1,3}) = \textbf{-3}$
\[
det(M_{1,3})=det(\begin{pmatrix}2 & -5\\-3 & 6\end{pmatrix}) = -3
\]
- $C_{2,3} = (-1)^{5} * det(M_{2,3}) = -det(M_{2,3}) = \textbf{-6}$
\[
det(M_{2,3})=det(\begin{pmatrix} -1 & 4 \\ -3 & 6 \end{pmatrix}) = 6
\]
- $C_{3,3} = (-1)^{6} * det(M_{3,3}) = det(M_{3,3}) = \textbf{-3}$
\[
det(M_{2,3})=det(\begin{pmatrix} -1 & 4 \\ 2 & -5\end{pmatrix}) = -3
\]
We can then calculate the determinant:
\[
\begin{split}
det(M)
= -7 * C_{1,3} + 8 * C_{2,3} + -9 * C_{3,3} \\
= -7 * -3 + 8 * -6 + -9 * -3 \\
= 0 \\
\end{split}
\]