###### Activity3.4.2

We will use the diagram to find the determinant of some simple $$2\times2$$ matrices.

DETERMINANTS

1. 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?

2. 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?

3. 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?

4. 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?

5. 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?

6. 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?

7. 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 $$\det B\text{?}$$

in-context