Let A, B, and C be real 2×2 matrices and denote A⋅B−B⋅A by [A, B]. Prove that: [[A,B], C]+[[B, C], A]+[[C, A], B]=0

Pertanyaan

Let begin mathsize 14px style A end stylebegin mathsize 14px style B end style, and begin mathsize 14px style C end style be real begin mathsize 14px style 2 cross times 2 end style matrices and denote begin mathsize 14px style A times B minus B times A end style by begin mathsize 14px style left square bracket A comma space B right square bracket end style. Prove that:

begin mathsize 14px style open square brackets left square bracket A comma B right square bracket comma space C close square brackets plus open square brackets left square bracket B comma space C right square bracket comma space A close square brackets plus open square brackets left square bracket C comma space A right square bracket comma space B close square brackets equals 0 end style  

A. Septianingsih

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Pembahasan

rightwards double arrow open square brackets open square brackets straight A comma straight B close square brackets comma straight C close square brackets plus open square brackets open square brackets straight B comma straight C close square brackets comma straight A close square brackets plus open square brackets open square brackets straight C comma straight A close square brackets comma straight B close square brackets equals 0 rightwards double arrow open square brackets open square brackets straight A. straight B minus straight B. straight A close square brackets comma straight C close square brackets plus open square brackets open square brackets straight B. straight C minus straight C. straight B close square brackets comma straight A close square brackets plus open square brackets open square brackets straight C. straight A minus straight A. straight C close square brackets comma straight B close square brackets equals 0 rightwards double arrow open square brackets open parentheses straight A. straight B minus straight B. straight A close parentheses. straight C minus straight C. open parentheses straight A. straight B minus straight B. straight A close parentheses close square brackets plus open square brackets open parentheses straight B. straight C minus straight C. straight B close parentheses. straight A minus straight A. open parentheses straight B. straight C minus straight C. straight B close parentheses close square brackets plus open square brackets open parentheses straight C. straight A minus straight A. straight C close parentheses. straight B minus straight B. open parentheses straight C. straight A minus straight A. straight C close parentheses close square brackets equals 0 rightwards double arrow ABC minus BAC minus CAB plus CBA plus BCA minus CBA minus ABC plus ACB plus CAB minus ACB minus BCA plus BAC equals 0 space left parenthesis terbukti right parenthesis 

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Pertanyaan serupa

Let A, B, and C be real 2×2 matrices and denote A⋅B−B⋅A by [A, B]. Prove that: [A, A]=0

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