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Given the two-by-two matrices π΄, which is equal to eight, negative three, one, negative two, and π΅, which is equal to eight, negative three, one, negative two, is π΄π΅ equal to π΅π΄?
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So weβve been given two two-by-two matrices.
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And weβre asked whether π΄π΅ is equal to π΅π΄.
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So weβre being asked whether we get the same result when we multiply the two matrices together in different orders.
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In general, we know that matrix multiplication is not commutative, which means that we get a different result if we multiply the matrices together in a different order.
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In fact, unless the two matrices are each square matrices of the same order, it wonβt even be possible to find both products.
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Matrix multiplication can be commutative under special circumstances.
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For example, if both matrices are diagonal matrices of the same order, meaning that they are square matrices.
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And all of the elements that arenβt on the leading diagonal are equal to zero.
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Matrix multiplication will also be commutative if one matrix is the identity matrix.
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And the other is a square matrix of the same order.
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Now, neither of these conditions apply here.
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But if we look at our two matrices π΄ and π΅, we see that they do have another relationship.
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π΄ and π΅ are each two-by-two matrices.
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So they have the same order.
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But more importantly, each of their elements in corresponding positions are the same.
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In this case then, matrix π΄ is entirely equal to matrix π΅.
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For this reason, it doesnβt matter in which order we multiply these two matrices together.
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The product π΄π΅ will be equal to the product π΅π΄.
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And in fact, theyβll both be equal to π΄ squared or indeed π΅ squared.
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We could of course confirm this by multiplying the matrices together longhand and checking that the two products do indeed give the same result.
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But there is no need.
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Once we spotted that the two matrices are identical, we know that their product will be the same regardless of which way around we find it.
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So we can conclude that, for these two particular matrices π΄ and π΅, but not in general, π΄π΅ is equal to π΅π΄.