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.
Activity3.1.2.
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.
Proposition3.1.1.
Let \(E\) be an elementary matrix and \(A\) be an \(n \times n\) matrix. Then
