Beyond the Basics

67.
1. Yes. By the definition of inverse, if $A B = I,$ $A B = I,$ then $B A = I$ $B A = I$ and A and B are inverses.
2. ${(A^{- 1})}^{- 1} = A$ ${(A^{- 1})}^{- 1} = A$
75. If A is invertible, then $I = A^{- 1} A = A^{- 1} (A B) = (A^{- 1} A) B = I B = B .$ $I = A^{- 1} A = A^{- 1} (A B) = (A^{- 1} A) B = I B = B .$
77.
1. $A^{- 1} = [\begin{array}{r} 3 & - 4 \\ - 2 & 3 \end{array}]$ $A^{- 1} = [\begin{array}{r} 3 & - 4 \\ - 2 & 3 \end{array}]$
79. ${(1, 2, 1, 2)}$ ${(1, 2, 1, 2)}$
81.
1. $A = [\begin{array}{r} 8 & 10 \\ 6 & 8 \end{array}]$ $A = [\begin{array}{r} 8 & 10 \\ 6 & 8 \end{array}]$
2. $A = [\begin{array}{r} - 1 & 7 \\ - 3 & 17 \end{array}]$ $A = [\begin{array}{r} - 1 & 7 \\ - 3 & 17 \end{array}]$
3. $A = [\begin{array}{r} \frac{3}{2} & \frac{1}{2} \\ \frac{5}{2} & \frac{3}{2} \end{array}]$ $A = [\begin{array}{r} \frac{3}{2} & \frac{1}{2} \\ \frac{5}{2} & \frac{3}{2} \end{array}]$
4. $A = [\begin{array}{r} - 1 & - 2 \\ \frac{5}{2} & 4 \end{array}]$ $A = [\begin{array}{r} - 1 & - 2 \\ \frac{5}{2} & 4 \end{array}]$

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