Pertanyaan

Misal A adalah matriks berordo ( 2 × 2 ) dan A − 1 menyatakan invers dari A, ∣ A ∣ menyatakan determinan A, dan k adalah sebuah konstanta. Buktikan bahwa determinan dari ( k A ) − 1 = k 2 ∣ A ∣ 1 .

Misal A adalah matriks berordo $(2 \times 2)$ dan $A^{- 1}$ menyatakan invers dari A, $∣ A ∣$ menyatakan determinan A, dan k adalah sebuah konstanta. Buktikan bahwa determinan dari $(k A)^{- 1} = \frac{1}{k ^{2} ∣ A ∣}$ .

H. Janatu

Master Teacher

Mahasiswa/Alumni Universitas Riau

Jawaban terverifikasi

Jawaban

terbukti bahwa .

terbukti bahwa $open vertical bar open parentheses k A close parentheses to the power of negative 1 end exponent close vertical bar equals fraction numerator 1 over denominator k squared open vertical bar A close vertical bar end fraction$ .

Pembahasan

Ingat rumus determinan dan invers matriks berikut: 1. 2. Akan dibuktikan bahwa Jadi, terbukti bahwa .

Ingat rumus determinan dan invers matriks berikut:

1. A equals open parentheses table row a b row c d end table close parentheses rightwards arrow open vertical bar A close vertical bar equals a times d minus b times c

2. $A equals open parentheses table row a b row c d end table close parentheses rightwards arrow A to the power of negative 1 end exponent equals fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses$

Akan dibuktikan bahwa $open vertical bar open parentheses k A close parentheses to the power of negative 1 end exponent close vertical bar equals fraction numerator 1 over denominator k squared open vertical bar A close vertical bar end fraction$

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell k A end cell equals cell k open parentheses table row a b row c d end table close parentheses end cell row blank equals cell open parentheses table row cell k a end cell cell k b end cell row cell k c end cell cell k d end cell end table close parentheses end cell row cell open parentheses k A close parentheses to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator k a times k d minus k b times k c end fraction open parentheses table row cell k d end cell cell negative k b end cell row cell negative k c end cell cell k a end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator k squared left parenthesis a d minus b c right parenthesis end fraction open parentheses table row cell k d end cell cell negative k b end cell row cell negative k c end cell cell k a end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator k d over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator negative k b over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell row cell fraction numerator negative k c over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator k a over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator d over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator negative b over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell row cell fraction numerator negative c over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell cell fraction numerator a over denominator k left parenthesis a d minus b c right parenthesis end fraction end cell end table close parentheses end cell row cell open vertical bar open parentheses k A close parentheses to the power of negative 1 end exponent close vertical bar end cell equals cell fraction numerator d over denominator k left parenthesis a d minus b c right parenthesis end fraction times fraction numerator a over denominator k left parenthesis a d minus b c right parenthesis end fraction minus open parentheses fraction numerator negative b over denominator k left parenthesis a d minus b c right parenthesis end fraction close parentheses open parentheses fraction numerator negative c over denominator k left parenthesis a d minus b c right parenthesis end fraction close parentheses end cell row blank equals cell fraction numerator a d over denominator k squared left parenthesis a d minus b c right parenthesis squared end fraction minus fraction numerator b c over denominator k squared left parenthesis a d minus b c right parenthesis squared end fraction end cell row blank equals cell fraction numerator a d minus b c over denominator k squared left parenthesis a d minus b c right parenthesis squared end fraction end cell row blank equals cell fraction numerator 1 over denominator k squared left parenthesis a d minus b c right parenthesis end fraction end cell row blank equals cell fraction numerator 1 over denominator k squared open vertical bar A close vertical bar end fraction end cell end table end style$

Jadi, terbukti bahwa $open vertical bar open parentheses k A close parentheses to the power of negative 1 end exponent close vertical bar equals fraction numerator 1 over denominator k squared open vertical bar A close vertical bar end fraction$ .

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