Activity 3.4.2
We will use the diagram to find the determinant of some simple \(2\times2\) matrices.
DETERMINANTS
Use the diagram to find the determinant of the matrix \(\left[\begin{array}{rr} \frac12 \amp 0 \\ 0 \amp 2 \end{array}\right]\text{.}\) What is the geometric effect of the matrix transformation defined by this matrix. What does this lead you to believe is generally true about the determinant of a diagonal matrix?
Use the diagram to find the determinant of the matrix \(\left[\begin{array}{rr} 0 \amp 1 \\ 1 \amp 0 \\ \end{array}\right]\text{.}\) What is the geometric effect of the matrix transformation defined by this matrix?
Use the diagram to find the determinant of the matrix \(\left[\begin{array}{rr} 2 \amp 1 \\ 0 \amp 1 \\ \end{array}\right]\text{.}\) What is the geometric effect of the matrix transformation defined by this matrix?
What do you notice about the determinant of any matrix of the form \(\left[\begin{array}{rr} 2 \amp k \\ 0 \amp 1 \\ \end{array}\right]\text{?}\) What does this say about the determinant of an upper triangular matrix?
Use the diagram to find the determinant of the matrix \(\left[\begin{array}{rr} 2 \amp 0 \\ 1 \amp 1 \\ \end{array}\right]\text{.}\) When we change the entry in the lower left corner, what is the effect on the determinant? What does this say about the determinant of a lower triangular matrix?
Use the diagram to find the determinant of the matrix \(\left[\begin{array}{rr} 1 \amp 1 \\ 2 \amp 2 \\ \end{array}\right]\text{.}\) What is the geometric effect of the matrix transformation defined by this matrix? In general, what is the determinant of a matrix whose columns are linearly dependent?

Consider the matrices
\begin{equation*} A = \left[\begin{array}{rr} 2 \amp 1 \\ 2 \amp 1 \\ \end{array}\right], B = \left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 2 \\ \end{array}\right]\text{.} \end{equation*}Use the diagram to find the determinants of \(A\text{,}\) \(B\text{,}\) and \(AB\text{.}\) What does this suggest is generally true about the relationship of \(\det(AB)\) to \(\det A\) and \(\det B\text{?}\)