Pertanyaan

Untuk A , B , dan X matriks berordo 2 × 2 . Buktikan bahwa: ( A B ) − 1 = B − 1 ⋅ A − 1

Untuk $A$ , $B$ , dan $X$ matriks berordo $2 \times 2$ .

Buktikan bahwa: $(A B)^{- 1} = B^{- 1} \cdot A^{- 1}$

A. Septianingsih

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Pembahasan

Misal space straight A equals open parentheses table row straight a straight b row straight c straight d end table close parentheses space dan space straight B equals open parentheses table row straight e straight f row straight g straight h end table close parentheses

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses A B close parentheses to the power of negative 1 end exponent end cell equals cell B to the power of negative 1 end exponent. A to the power of negative 1 end exponent end cell row cell open square brackets open parentheses table row a b row c d end table close parentheses. open parentheses table row e f row g h end table close parentheses close square brackets to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator e h minus f g end fraction open parentheses table row h cell negative f end cell row cell negative g end cell e end table close parentheses. fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses end cell row cell open square brackets open parentheses table row cell a e plus b g end cell cell a f plus b h end cell row cell c e plus d g end cell cell c f plus d h end cell end table close parentheses close square brackets to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator open parentheses e h minus f g close parentheses. open parentheses a d minus b c close parentheses end fraction open parentheses table row cell d h plus c f end cell cell negative b h minus a f end cell row cell negative d g minus c e end cell cell b g plus a e end cell end table close parentheses end cell row blank blank cell fraction numerator 1 over denominator open parentheses a e plus b g close parentheses open parentheses c f plus d h close parentheses minus open parentheses c e plus d g close parentheses open parentheses a f plus b h close parentheses end fraction open parentheses table row cell c f plus d h end cell cell negative open parentheses a f plus b h close parentheses end cell row cell negative open parentheses c e plus d f close parentheses end cell cell a e plus b g end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator open parentheses e h minus f g close parentheses. open parentheses a d minus b c close parentheses end fraction open parentheses table row cell d h plus c f end cell cell negative b h minus a f end cell row cell negative d g minus c e end cell cell b g plus a e end cell end table close parentheses end cell row blank blank cell fraction numerator 1 over denominator a e c f plus a e d h plus b g c f plus b g d h minus a e c f minus c e b h minus d g a f minus d g b h end fraction open parentheses table row cell c f plus d h end cell cell negative open parentheses a f plus b h close parentheses end cell row cell negative open parentheses c e plus d f close parentheses end cell cell a e plus b g end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator a d e h minus b c e h minus a d f g plus b c f g end fraction open parentheses table row cell d h plus c f end cell cell negative b h minus a f end cell row cell negative d g minus c e end cell cell b g plus a e end cell end table close parentheses end cell row blank blank cell fraction numerator 1 over denominator a d e h minus b c e h minus a d f g plus b c f g end fraction open parentheses table row cell c f plus d h end cell cell negative open parentheses a f plus b h close parentheses end cell row cell negative open parentheses c e plus d f close parentheses end cell cell a e plus b g end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator a d e h minus b c e h minus a d f g plus b c f g end fraction open parentheses table row cell c f plus d h end cell cell negative open parentheses a f plus b h close parentheses end cell row cell negative open parentheses c e plus d f close parentheses end cell cell a e plus b g end cell end table close parentheses space end cell row blank blank cell left parenthesis T E R B U K T I right parenthesis end cell row blank blank blank end table$