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Matrix Calculations A subspace of linear algebra

Chrissy Safranski

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1. Inverse and row operations, diagonal matrix.

Use row reducing to calculating the inverses of the following two matrices. What do you notice about the matrices \(A^{-1}\) and \(B\text{?}\) What about \(A\) and \(B^{-1}\text{?}\)
\begin{align*} A=\left[\begin{array}{rrrr} 2 \amp 0 \amp 0 \amp 0 \\ 0 \amp -3 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \amp -\frac{2}{3} \end{array}\right] \amp \amp \amp \amp \amp \amp \amp \amp B=\left[\begin{array}{rrrr} \frac{1}{2} \amp 0 \amp 0 \amp 0 \\ 0 \amp -\frac{1}{3} \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \amp -\frac{3}{2} \end{array}\right] \amp \amp \amp \amp \amp \amp \amp \amp \end{align*}
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2. Inverse and row operations, lower triangular matrix.

(a)
Use row reducing to calculate the inverse of the matrix below. Be sure to write down which row operations you did.
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\(C=\left[\begin{array}{rrr} 2 \amp 0 \amp 0 \\ 3 \amp -3 \amp 0 \\ -2 \amp 1 \amp 1 \end{array}\right]\)
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(b)
Even though this is not our usual Gauss-Jordan algorithm, perform the following row operations in order on the matrix \(D\text{.}\) What does the result tell you about \(D^{-1}\text{?}\) What do you notice about the row operations compared to the row operations done on \(C\text{?}\)
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\begin{align*} R_2+R_3\amp \rightarrow R_3\\ -3R_2\amp \rightarrow R_2\\ -2R_1+R_3\amp \rightarrow R_3\\ 3R_1+R_2\amp \rightarrow R_2\\ 2R_1\amp \rightarrow R_1 \end{align*}
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\(D=\left[\begin{array}{rrr|rrr}\frac{1}{2} \amp 0 \amp 0 \amp 1 \amp 0 \amp 0 \\ \frac{1}{2} \amp -\frac{1}{3} \amp 0 \amp 0 \amp 1 \amp 0\\ \frac{1}{2} \amp \frac{1}{3} \amp 1\amp 0 \amp 0 \amp 1\end{array}\right]\)
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3. Investigating properties of the inverse.

For this problem, use Sage or another matrix inverse calculator. Note that once you’ve defined the matrix \(A\) in Sage, Sage can find the inverse without needing to augment and row reduce using A.inverse() or A^-1.
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Consider the square matrices
\begin{align*} A = \begin{bmatrix} 1 \amp 0 \amp 2 \\ 2 \amp 2 \amp 1 \\ 1 \amp 1 \amp 1 \\ \end{bmatrix} \amp \amp \amp \amp \amp \amp B = \left[\begin{array}{rrr} 1 \amp -2 \amp -1 \\ -1 \amp 5 \amp 6 \\ 5 \amp -4 \amp 6 \\ \end{array}\right] \end{align*}
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(b)
Show that \(AB\) is invertible by finding a matrix \(X\) such that \((AB)X=I\text{.}\)
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(c)
Use Sage to calculate \(AB\text{,}\) \((AB)^{-1}\text{,}\) \(A^{-1}B^{-1}\text{,}\) and \(B^{-1}A^{-1}\text{.}\) What do you notice?
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(d)
Consider the product
\begin{equation*} (EF)(E^{-1}F^{-1}) \end{equation*}
and use associativity to regroup the products so that the middle two terms are multiplied first. Do the same for
\begin{equation*} (F^{-1}E^{-1})(EF)\text{.} \end{equation*}
What do you get?
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(e)
What is the inverse of \(B^2\) in terms of \(B^{-1}\text{?}\) What about the inverse of \(B^5\text{?}\) Use Sage to verify your conjectures.
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(f)
Calculate \(A+B\text{,}\) \((A+B)^{-1}\) and \(A^{-1}+B^{-1}\text{.}\) What do you observe?
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