Activity 3.1.3

We'll begin by considering the square matrix $$A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 2 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}.$$

1. Describe the solution space to the equation $A\xvec = \threevec343$ by augmenting $A$ and finding the reduced row echelon form.

2. Using this proposition, explain why $A$ is invertible and find its inverse.

3. Now use the inverse to solve the equation $A\xvec = \threevec343$ and verify that your result agrees with what you found in part a.

4. If you have defined a matrix B in Sage, you can find it's inverse as B.inverse() or B^-1. Use Sage to find the inverse of the matrix $$B = \left[\begin{array}{rrr} 1 & -2 & -1 \\ -1 & 5 & 6 \\ 5 & -4 & 6 \\ \end{array}\right]$$ and use it to solve the equation $B\xvec = \threevec83{36}$.

5. If $A$ and $B$ are the two matrices defined in this activity, find their product $AB$ and verify that it is invertible.

6. Compute the products $A^{-1}B^{-1}$ and $B^{-1}A^{-1}$. Which one agrees with $(AB)^{-1}$?

7. Explain your finding by considering the product $$(AB)(B^{-1}A^{-1})$$ and using associativity to regroup the products so that the middle two terms are multiplied first.