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*}
Section 3.1 Determinant
Subsection 3.1.1 Fast way to get the determinant
Activity 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*}
Theorem 3.1.1.
Let \(A\) be an \(n \times n\) matrix. \(\det(A)\neq 0\) if and only if \(A\) is invertible.
Subsection 3.1.2 The slow one
If you would like to learn more! In next activity, we perform three elementary row operations to yield matrices, and then find the determinants, respectively.
Activity 3.1.3.
Swap the rows 2 and 3 to yields the matrix \(B=\left[\begin{array}{rrrr}
2 \amp 1 \amp -2 \amp -3 \\
-3 \amp 4 \amp 1 \amp 2\\
3 \amp 0 \amp -1 \amp -2 \\
1 \amp 3 \amp 3 \amp -1 \\
\end{array}\right]\text{.}\) \(C= \left[\begin{array}{rrrr}
2 \amp 1 \amp -2 \amp -3 \\
9 \amp 0 \amp -3 \amp -6 \\
-3 \amp 4 \amp 1 \amp 2\\
1 \amp 3 \amp 3 \amp -1 \\
\end{array}\right]\) is obtained by scaling row 2 of the matrix \(A\) by 3. Let \(D = \left[\begin{array}{rrrr}
2 \amp 1 \amp -2 \amp -3 \\
3 \amp 0 \amp -1 \amp -2 \\
-3 \amp 4 \amp 1 \amp 2\\
7 \amp 3 \amp 1 \amp -5 \\
\end{array}\right]\) is obtained by adding 2 times of row 2 of the matrix \(A\) to row 4.
Find the determinant of \(B\text{,}\) and state what you find.
Find the determinant of \(C\text{,}\) and state what you find.
Find the determinant of \(D\text{,}\) and state what you find.
Proposition 3.1.2.
Let \(E\) be an elementary matrix and \(A\) be an \(n \times n\) matrix. Then
\begin{equation*}
\det(EA)=\det(E)\det(A)
\end{equation*}
It follows from Theorem 2.4.2 that
Theorem 3.1.3.
Let \(A\) be an \(n \times n\) matrix. \(\det(A)\neq 0\) if and only if \(A\) is invertible.
