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 \(\det A^{-1}\text{?}\)

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