A pair of vectors \(\vvec_1\) and \(\vvec_2\) is called positively oriented if the angle, measured in the counterclockwise direction, from \(\vvec_1\) to \(\vvec_2\) is less than \(180^\circ\text{;}\) we say the pair is negatively oriented if it is more than \(180^\circ\text{.}\)
Definition3.4.0.1.
Suppose a \(2\times2\) matrix \(A\) has columns \(\vvec_1\) and \(\vvec_2\text{.}\) If the pair of vectors is positively oriented, then the determinant of \(A\text{,}\) denoted \(\det A\text{,}\) is the area of the parallelogram formed by \(\vvec_1\) and \(\vvec_2\text{.}\) If the pair is negatively oriented, then \(\det A\) is minus the area of the parallelogram.
For the next several activities, We will use geometry to find the determinant of some simple \(2\times2\) matrices.
Activity3.4.0.1.
Use the diagram in Activity 3.4.2 9 of Undertanding Linear Algebra or your own sketch of the appropriate parallelogram to find the determinant.
(a)
Find the determinant of the matrix \(\left[\begin{array}{rr} -\frac12 \amp 0 \\ 0 \amp 2
\end{array}\right]\text{.}\)
(b)
What is the geometric effect of the matrix transformation defined by this matrix?
(c)
What does this lead you to believe is generally true about the determinant of a diagonal matrix?
Activity3.4.0.2.
Use the diagram in Activity 3.4.2 10 of Undertanding Linear Algebra or your own sketch of the appropriate parallelogram to find the determinant.
(a)
Find the determinant of the matrix \(\left[\begin{array}{rr} 0 \amp 1 \\
1 \amp 0 \\
\end{array}\right]\text{.}\)
(b)
What is the geometric effect of the matrix transformation defined by this matrix?
Activity3.4.0.3.
Use the diagram in Activity 3.4.2 11 of Undertanding Linear Algebra or your own sketch of the appropriate parallelogram to find the determinant.
(a)
Find the determinant of the matrix \(\left[\begin{array}{rr} 2 \amp 1 \\
0 \amp 1 \\
\end{array}\right]\text{.}\)
(b)
What is the geometric effect of the matrix transformation defined by this matrix?
(c)
What do you notice about the determinant of any matrix of the form \(\left[\begin{array}{rr} 2 \amp k \\
0 \amp 1 \\
\end{array}\right]\text{?}\)
(d)
What does this say about the determinant of an upper triangular matrix?
Activity3.4.0.4.
Use the diagram in Activity 3.4.2 12 of Undertanding Linear Algebra or your own sketch of the appropriate parallelogram to find the determinant.
(a)
Find the determinant of the matrix \(\left[\begin{array}{rr} 2 \amp 0 \\
1 \amp 1 \\
\end{array}\right]\text{.}\)
(b)
When we change the entry in the lower left corner, what is the effect on the determinant?
(c)
What does this say about the determinant of a lower triangular matrix?
Activity3.4.0.5.
Use the diagram in Activity 3.4.2 13 of Undertanding Linear Algebra or your own sketch of the appropriate parallelogram to find the determinant.
(a)
Find the determinant of the matrix \(\left[\begin{array}{rr} 1 \amp -1 \\
-2 \amp 2 \\
\end{array}\right]\text{.}\)
(b)
What is the geometric effect of the matrix transformation defined by this matrix?
(c)
In general, what is the determinant of a matrix whose columns are linearly dependent?
Use the diagram in Activity 3.4.2 14 of Undertanding Linear Algebra or your own sketch of the appropriate parallelogram to find the determinants of \(A\text{,}\)\(B\text{,}\) and \(AB\text{.}\)
(b)
What does this suggest is generally true about the relationship of \(\det(AB)\) to \(\det A\) and \(\det B\text{?}\)
The next several activities will investigate the connection between the determinant of a matrix and its invertibility using Gaussian elimination.
Which of the matrices \(U_1\) and \(U_2\) are invertible?
(b)
Use our earlier observation that the determinant of an upper triangular matrix is the product of its diagonal entries to find \(\det U_1\) and \(\det U_2\text{.}\)
(c)
Explain why an upper triangular matrix is invertible if and only if its determinant is not zero.
to arrive at an upper triangular matrix \(U\text{.}\) What is the relationship between \(\det A\) and \(\det U\text{?}\)
(d)
Since \(U\) is upper triangular, we can compute its determinant, which allows us to find \(\det A\text{.}\) What is \(\det A\text{?}\) Is \(A\) invertible?
Perform a sequence of row operations to find an upper triangular matrix \(U\) that is row equivalent to \(A\text{.}\) Use this to determine \(\det A\text{.}\)
(b)
Is the matrix \(A\) invertible?
Activity3.4.0.10.
Suppose we apply a sequence of row operations on a matrix \(A\) to obtain \(A'\text{.}\)
(a)
Explain why \(\det A
\neq 0\) if and only if \(\det A' \neq 0\text{.}\)
(b)
Explain why an \(n\times n\) matrix \(A\) is invertible if and only if \(\det A \neq 0\text{.}\)
Activity3.4.0.11.
If \(A\) is an invertible matrix with \(\det A =
-3\text{,}\) what is \(\det A^{-1}\text{?}\)
Another way to calculate the determinant is through cofactor expansion. We will explore cofactor expansions through some examples.
Activity3.4.0.12.
Using a cofactor expansion, show that the determinant of the following matrix
