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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

Let , and  be real  matrices and denote  by . Prove that:

  

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A. Septianingsih

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

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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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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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