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Prove that 2x2 matrix multiplication is associative.Solution .pdf

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Prove that 2x2 matrix multiplication is associative. Solution Let there be three matrices A,B and C Let the entries of the matrices be denoted by a11, a12, a21, a22 for A, etc. So the ij entry of AB is: ai1 b1j + ai2 b2j The ij entry of (AB)C is then: (ai1 b11 + ai2 b21)c1j + (ai1 b12 + ai2 b22)c2j On the other hand, the ij entry of BC is: bi1 c1j + bi2 c2j So the ij entry of A(BC) is: ai1(b11 c1j + b12 c2j) + ai2(b21 c1j + b22 c2j) These are the same expressions for (AB)C and A(BC)..

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Prove that 2x2 matrix multiplication is associative. Solution Let there be three matrices A,B and C Let the entries of the matrices be denoted by a11, a12, a21, a22 for A, etc. So the ij entry of AB is: ai1 b1j + ai2 b2j The ij entry of (AB)C is then: (ai1 b11 + ai2 b21)c1j + (ai1 b12 + ai2 b22)c2j On the other hand, the ij entry of BC is: bi1 c1j + bi2 c2j So the ij entry of A(BC) is: ai1(b11 c1j + b12 c2j) + ai2(b21 c1j + b22 c2j) These are the same expressions for (AB)C and A(BC).

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Prove that 2x2 matrix multiplication is associative.Solution .pdf

  • 1. Prove that 2x2 matrix multiplication is associative. Solution Let there be three matrices A,B and C Let the entries of the matrices be denoted by a11, a12, a21, a22 for A, etc. So the ij entry of AB is: ai1 b1j + ai2 b2j The ij entry of (AB)C is then: (ai1 b11 + ai2 b21)c1j + (ai1 b12 + ai2 b22)c2j On the other hand, the ij entry of BC is: bi1 c1j + bi2 c2j So the ij entry of A(BC) is: ai1(b11 c1j + b12 c2j) + ai2(b21 c1j + b22 c2j) These are the same expressions for (AB)C and A(BC).