Activity3.4.3

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

1. 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 $$\det U_2\text{.}$$

2. Explain why an upper triangular matrix is invertible if and only if its determinant is not zero.

3. 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 $$\det A_1\text{?}$$

4. 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 $$\det A_2\text{?}$$

5. 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 $$\det U\text{?}$$

6. 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?

7. 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?

8. 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 $$\det A' \neq 0\text{.}$$

9. Explain why an $$n\times n$$ matrix $$A$$ is invertible if and only if $$\det A \neq 0\text{.}$$

10. If $$A$$ is an invertible matrix with $$\det A = -3\text{,}$$ what is $$\det A^{-1}\text{?}$$

in-context