Activity 3.4.3
We will investigate the connection between the determinant of a matrix and its invertibility using Gaussian elimination.

Consider the two upper triangular matrices
\begin{equation*} U_1 = \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 0 \amp 2 \amp 4 \\ 0 \amp 0 \amp 2 \\ \end{array}\right], U_2 = \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 0 \amp 2 \amp 4 \\ 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}Which of the matrices \(U_1\) and \(U_2\) are invertible? 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 \(\det U_2\text{.}\)
Explain why an upper triangular matrix is invertible if and only if its determinant is not zero.

Let's now consider the matrix
\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 2 \amp 2 \amp 6 \\ 3 \amp 1 \amp 10 \\ \end{array}\right] \end{equation*}and start the Gaussian elimination process. We begin with a row replacement operation
\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 2 \amp 2 \amp 6 \\ 3 \amp 1 \amp 10 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 0 \amp 0 \amp 2 \\ 3 \amp 1 \amp 10 \\ \end{array}\right] = A_1\text{.} \end{equation*}What is the relationship between \(\det A\) and \(\det A_1\text{?}\)

Next we perform another row replacement operation:
\begin{equation*} A_1= \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 0 \amp 0 \amp 2 \\ 3 \amp 1 \amp 10 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 0 \amp 0 \amp 2 \\ 0 \amp 2 \amp 4 \\ \end{array}\right] = A_2\text{.} \end{equation*}What is the relationship between \(\det A\) and \(\det A_2\text{?}\)

Finally, we perform an interchange:
\begin{equation*} A_2 = \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 0 \amp 0 \amp 2 \\ 0 \amp 2 \amp 4 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 \amp 1 \amp 2 \\ 0 \amp 2 \amp 4 \\ 0 \amp 0 \amp 2 \\ \end{array}\right] = U \end{equation*}to arrive at an upper triangular matrix \(U\text{.}\) What is the relationship between \(\det A\) and \(\det U\text{?}\)
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?

Now consider the matrix
\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 1 \amp 3 \\ 0 \amp 2 \amp 2 \\ 2 \amp 1 \amp 3 \\ \end{array}\right]\text{.} \end{equation*}Perform a sequence of row operations to find an upper triangular matrix \(U\) that is row equivalent to \(A\text{.}\) Use this to determine \(\det A\text{.}\) Is the matrix \(A\) invertible?
Suppose we apply a sequence of row operations on a matrix \(A\) to obtain \(A'\text{.}\) Explain why \(\det A \neq 0\) if and only if \(\det A' \neq 0\text{.}\)
Explain why an \(n\times n\) matrix \(A\) is invertible if and only if \(\det A \neq 0\text{.}\)
If \(A\) is an invertible matrix with \(\det A = 3\text{,}\) what is \(\det A^{1}\text{?}\)