## Elementary Linear Algebra 7th Edition

(a) $$(AB)^{-1}=\left[ \begin {array}{cc} {-\frac{4}{77}}& {\frac{9}{77}}\\ {-\frac{9}{77}} & {\frac{1}{77}} \end {array} \right] .$$ (b) $$(A^T)^{-1}= \left[ \begin {array}{cc} {-\frac{2}{7}}& {\frac{3}{7}}\\ {\frac{1}{7}} & {\frac{2}{7}} \end {array} \right].$$ (c) $$(2A)^{-1}= \left[ \begin {array}{cc} {-\frac{2}{14}}& {\frac{1}{14}}\\ {\frac{3}{14}} & {\frac{2}{14}} \end {array} \right].$$
(a) $$(AB)^{-1}=B^{-1}A^{-1}=\left[ \begin {array}{cc} {\frac{5}{11}}& {\frac{2}{11}}\\ {\frac{3}{11}} & {-\frac{1}{11}} \end {array} \right]\left[ \begin {array}{cc} {-\frac{2}{7}}& {\frac{1}{7}}\\ {\frac{3}{7}} & {\frac{2}{7}} \end {array} \right] =\left[ \begin {array}{cc} {-\frac{4}{77}}& {\frac{9}{77}}\\ {-\frac{9}{77}} & {\frac{1}{77}} \end {array} \right] .$$ (b) $$(A^T)^{-1}=(A^{-1})^T=\left[ \begin {array}{cc} {-\frac{2}{7}}& {\frac{3}{7}}\\ {\frac{1}{7}} & {\frac{2}{7}} \end {array} \right].$$ (c) $$(2A)^{-1}=\frac{1}{2}A^{-1}=\frac{1}{2}\left[ \begin {array}{cc} {-\frac{2}{7}}& {\frac{1}{7}}\\ {\frac{3}{7}} & {\frac{2}{7}} \end {array} \right]=\left[ \begin {array}{cc} {-\frac{2}{14}}& {\frac{1}{14}}\\ {\frac{3}{14}} & {\frac{2}{14}} \end {array} \right].$$