Activity 3.4.3

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

1. Consider the two upper triangular matrices $$U_1 = \left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 2 & 4 \\ 0 & 0 & -2 \\ \end{array}\right],~~~ U_2 = \left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 2 & 4 \\ 0 & 0 & 0 \\ \end{array}\right].$$ Remembering this proposition, which of the matrices $U_1$ and $U_2$ are invertible? What are the determinants $\det(U_1)$ and $\det(U_2)$?

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 $$A = \left[\begin{array}{rrr} 1 & -1 & 2 \\ -2 & 2 & -6 \\ 3 & -1 & 10 \\ \end{array}\right]$$ and begin the Gaussian elimination process with a row replacement operation $$A = \left[\begin{array}{rrr} 1 & -1 & 2 \\ -2 & 2 & -6 \\ 3 & -1 & 10 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 0 & -2 \\ 3 & -1 & 10 \\ \end{array}\right] = A_1$$. What is the relationship between $\det(A)$ and $\det(A_1)$?

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

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

6. Since $U$ is upper triangular, we can compute its determinant, which allows us to find $\det(A)$. What is $\det(A)$? Is $A$ invertible?

7. Now consider the matrix $$A = \left[\begin{array}{rrr} 1 & -1 & 3 \\ 0 & 2 & -2 \\ 2 & 1 & 3 \\ \end{array}\right].$$ Perform a sequence of row operations to find an upper triangular matrix $U$ that is row equivalent to $A$. Use this to determine $\det(A)$ and whether $A$ is invertible.

8. Suppose we apply a sequence of row operations on a matrix $A$ to obtain $A'$. Explain why $\det(A) \neq 0$ if and only if $\det(A') \neq 0$.

9. Explain why an $n\times n$ matrix $A$ is invertible if and only if $\det(A) \neq 0$.