Objectives
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Practice calculating with elementary matrices, including multiplication and inverses.
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Observe and deepen understanding of connections between solutions of linear systems, invertibility, and elementary matrices.
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1.
Consider the matrix \(A=\left[\begin{array}{rrr} 1 \amp 0 \amp 2 \\ 0 \amp 1 \amp -1 \\ -3 \amp 0 \amp -7 \end{array}\right]\text{.}\) Note that performing Gauss-Jordan elimination requires performing in order
(a)
Calculate \(E_1\text{,}\) \(E_2\text{,}\) \(E_3\text{,}\) and \(E_4\) without using technology. What is the inverse of each matrix you found?
(b)
Use the sage cell below and modify it to use what you got for \(E_1\) and verify that \(E_1A\) performs \(3R_1+R_3 \rightarrow R_3\) on \(A\text{.}\)
(c)
Now similarly modify the sage cell below with what you found for \(E_2\text{,}\) \(E_3\text{,}\) and \(E_4\) and verify that \(E_4E_3E_2E_1A=I\text{.}\)
(d)
What do the calculations in the cell below tell us about \(A^{-1}\) and elementary matrices?
(e)
Use sage to verify your calculations of the inverses in the first part, and to verify that
\begin{equation*}
A=E_1^{-1}E_2^{-1}E_3^{-1}E_4^{-1}\text{.}
\end{equation*}
2.
The theorem containing various things equivalent to the property of being invertible is sometimes called “The Invertible Matrix Theorem”. But the star of the show is really the property of the reduced row echelon form of \(A\) being \(I\text{.}\) Let’s explore a bit how some of the various properties are related.
(a)
Suppose you know that \(A\) is an \(n\times n\) matrix whose reduced row echelon form is \(I\text{.}\) Set up the augmented matrix \(\big[A | I \big]\text{.}\) What happens after you row reduce this matrix? What does mean for the invertibility of \(A\text{?}\)
(b)
Suppose you know that \(A\) is an \(n\times n\) matrix whose reduced row echelon form is \(I\text{.}\) Now we know that performing a row operation is the same as multiplying by an elementary matrix. Performing a row operation on \(\big[A | I \big]\) has the same result as \(\big[E_1A | E_1I \big]\text{.}\) The reduced row echelon form of \(A\) being \(I\) means that there is a sequence of elementary matrices \(E_1, E_2, \ldots E_m\) such that \(E_m\cdots E_2 E_1A=I\text{.}\) What does this mean about \(E_m\cdots E_2 E_1\text{?}\)
(c)
Suppose you know that \(A\) is an invertible \(n\times n\) matrix and you have a linear system with \(A\) as its coefficient matrix. Then you can write \(AX=B\) where \(X\) is a column matrix of variables and \(B\) is your column matrix of constants. How many solutions are there for \(X\text{?}\) How do you know?
(d)
Suppose you know that \(A\) is an invertible \(n\times n\) matrix and that \(X\) is a matrix with \(n\) rows such that \(AX=\mathbf{0}\text{.}\) What does \(X\) have to equal? How do you know?
(e)
Suppose that \(A\) is an \(n\times n\) matrix and \(X\) is a column matrix with \(n\) rows. Now suppose that the only solution to \(AX=\mathbf{0}\) is that \(X=\mathbf{0}\text{.}\) (Why isn’t it possible that there are no solutions?) Set up \(\big[A | \mathbf{0} \big]\text{.}\) After row reducing to solve the system, how many free variables can there be? What does the reduced row echelon form have to look like?
