Question #245556

Given

A=(2224)andB=(2321).A=\left(\begin{array}{rr}-2&2\\2&-4\end{array}\right)\quad \textrm{and} \quad B=\left(\begin{array}{rr}2&3\\2&-1\end{array}\right).


Select the option(s) below which represent (AB1)1.\left(AB^{-1}\right)^{-1}.

  1. (3.52.51.50.5).\displaystyle \left(\begin{array}{rr}-3.5&-2.5\\-1.5&-0.5\end{array}\right).
  2. (3.52.51.50.5).\displaystyle \left(\begin{array}{rr}3.5&2.5\\1.5&0.5\end{array}\right).
  3. A-1B
  4. B-1A-1
  5. BA-1




1
Expert's answer
2021-10-05T11:00:24-0400

Let's find B1{B^{ - 1}} :

(23211001)I(23041011)(2301101414)3II(200114341414)(100118381414)\left( {\left. {\begin{matrix} 2&3\\ 2&{ - 1} \end{matrix}} \right|\begin{matrix} 1&0\\ 0&1 \end{matrix}} \right)\begin{matrix} {}\\ { - I} \end{matrix} \Leftrightarrow \left( {\left. {\begin{matrix} 2&3\\ 0&{ - 4} \end{matrix}} \right|\begin{matrix} 1&0\\ { - 1}&1 \end{matrix}} \right) \Leftrightarrow \left( {\left. {\begin{matrix} 2&3\\ 0&1 \end{matrix}} \right|\begin{matrix} 1&0\\ {\frac{1}{4}}&{ - \frac{1}{4}} \end{matrix}} \right)\begin{matrix} { - 3II}\\ {} \end{matrix} \Leftrightarrow \left( {\left. {\begin{matrix} 2&0\\ 0&1 \end{matrix}} \right|\begin{matrix} {\frac{1}{4}}&{\frac{3}{4}}\\ {\frac{1}{4}}&{ - \frac{1}{4}} \end{matrix}} \right) \Leftrightarrow \left( {\left. {\begin{matrix} 1&0\\ 0&1 \end{matrix}} \right|\begin{matrix} {\frac{1}{8}}&{\frac{3}{8}}\\ {\frac{1}{4}}&{ - \frac{1}{4}} \end{matrix}} \right)

Then

B1=(18381414){B^{ - 1}} = \left( {\begin{matrix} {\frac{1}{8}}&{\frac{3}{8}}\\ {\frac{1}{4}}&{ - \frac{1}{4}} \end{matrix}} \right)

AB1=(2224)(18381414)=(14+12341214134+1)=(14543474)A{B^{ - 1}} = \left( {\begin{matrix} { - 2}&2\\ 2&{ - 4} \end{matrix}} \right)\left( {\begin{matrix} {\frac{1}{8}}&{\frac{3}{8}}\\ {\frac{1}{4}}&{ - \frac{1}{4}} \end{matrix}} \right) = \left( {\begin{matrix} { - \frac{1}{4} + \frac{1}{2}}&{ - \frac{3}{4} - \frac{1}{2}}\\ {\frac{1}{4} - 1}&{\frac{3}{4} + 1} \end{matrix}} \right) = \left( {\begin{matrix} {\frac{1}{4}}&{ - \frac{5}{4}}\\ { - \frac{3}{4}}&{\frac{7}{4}} \end{matrix}} \right)

Let's find (AB1)1{\left( {A{B^{ - 1}}} \right)^{ - 1}} :

(AB1)1=(145434741001)(15374004)+3I(150840124)(15024031)52II(1002725231)(100172523212){\left( {A{B^{ - 1}}} \right)^{ - 1}} = \left( {\left. {\begin{matrix} {\frac{1}{4}}&{ - \frac{5}{4}}\\ { - \frac{3}{4}}&{\frac{7}{4}} \end{matrix}} \right|\begin{matrix} 1&0\\ 0&1 \end{matrix}} \right) \Leftrightarrow \left( {\left. {\begin{matrix} 1&{ - 5}\\ { - 3}&7 \end{matrix}} \right|\begin{matrix} 4&0\\ 0&4 \end{matrix}} \right)\begin{matrix} {}\\ { + 3I} \end{matrix} \Leftrightarrow \left( {\left. {\begin{matrix} 1&{ - 5}\\ 0&{ - 8} \end{matrix}} \right|\begin{matrix} 4&0\\ {12}&4 \end{matrix}} \right) \Leftrightarrow \left( {\left. {\begin{matrix} 1&{ - 5}\\ 0&{ - 2} \end{matrix}} \right|\begin{matrix} 4&0\\ 3&1 \end{matrix}} \right)\begin{matrix} { - \frac{5}{2}II}\\ {} \end{matrix} \Leftrightarrow \left( {\left. {\begin{matrix} 1&0\\ 0&{ - 2} \end{matrix}} \right|\begin{matrix} { - \frac{7}{2}}&{ - \frac{5}{2}}\\ 3&1 \end{matrix}} \right) \Leftrightarrow \left( {\left. {\begin{matrix} 1&0\\ 0&1 \end{matrix}} \right|\begin{matrix} { - \frac{7}{2}}&{ - \frac{5}{2}}\\ { - \frac{3}{2}}&{ - \frac{1}{2}} \end{matrix}} \right)

Then

(AB1)1=(3.52.51.50.5){\left( {A{B^{ - 1}}} \right)^{ - 1}} = \left( {\begin{matrix} { - 3.5}&{ - 2.5}\\ { - 1.5}&{ - 0.5} \end{matrix}} \right)

By the properties of the inverse matrix

(AB1)1=(B1)1A1=BA1{\left( {A{B^{ - 1}}} \right)^{ - 1}} = {\left( {{B^{ - 1}}} \right)^{ - 1}}{A^{ - 1}} = B{A^{ - 1}}

Answer: 1. (AB1)1=(3.52.51.50.5){\left( {A{B^{ - 1}}} \right)^{ - 1}} = \left( {\begin{matrix} { - 3.5}&{ - 2.5}\\ { - 1.5}&{ - 0.5} \end{matrix}} \right) , 5. BA1B{A^{ - 1}}




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