Q 1 . {\displaystyle Q1\mathrm {.} } 행렬 A = ( 3 2 1 1 ) {\displaystyle {A}\mathrm {=} \left({\begin{array}{ll}{3}&{2}\\{1}&{1}\end{array}}\right)} 에 대하여 A + A − 1 {\displaystyle {A}\mathrm {+} {A}^{\mathrm {-} {1}}} 의 모든 성분의 합을 구하라.
A = ( a b c d ) , A − 1 = 1 a d − b c ( d − b − c a ) {\displaystyle {A}\mathrm {=} \left({\begin{array}{ll}{a}&{b}\\{c}&{d}\end{array}}\right){\mathrm {,} }\;{A}^{\mathrm {-} {1}}\mathrm {=} {\frac {1}{{ad}\mathrm {-} {bc}}}\left({\begin{array}{ll}{d}&{\mathrm {-} {b}}\\{\mathrm {-} {c}}&{a}\end{array}}\right)}
A − 1 = ( 1 − 2 − 1 3 ) {\displaystyle {A}^{\mathrm {-} {1}}\mathrm {=} \mathrm {\left({\begin{array}{ll}{\;{1}}&{{-}{2}}\\{{-}{1}}&{\;{3}}\end{array}}\right)} }
A + A − 1 = ( 3 2 1 1 ) + ( 1 − 2 − 1 3 ) = ( 4 0 0 4 ) {\displaystyle {\begin{array}{ccc}{{A}\mathrm {+} {A}^{\mathrm {-} {1}}}&{\mathrm {=} }&{\mathrm {\left({\begin{array}{ll}{3}&{2}\\{1}&{1}\end{array}}\right)} \mathrm {+} \mathrm {\left({\begin{array}{ll}{1}&{{-}{2}}\\{{-}{1}}&{3}\end{array}}\right)} }\\{}&{\mathrm {=} }&{\mathrm {\left({\begin{array}{ll}{4}&{0}\\{0}&{4}\end{array}}\right)} }\end{array}}}
4 + 0 + 4 + 0 = 8 ∴ 8 {\displaystyle {\begin{aligned}&{{4}\mathrm {+} {0}\mathrm {+} {4}\mathrm {+} {0}\mathrm {=} {8}}\\&{\mathrm {\therefore } \;{8}}\end{aligned}}}
