WEBVTT
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Let π΄ equal the matrix one, two, three, four and π΅ equal the matrix one, two, one, π.
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Is it possible to choose a value for π so that π΄π΅ is equal to π΅π΄?
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If so, what is this value?
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So the first thing we need to do is remind ourselves how we multiply two two-by-two matrices.
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So letβs imagine weβve got the matrices π, π, π, π and π, π, π, β.
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Then the first element of the result, which will be a two-by-two matrix, is gonna be π multiplied by π, so the first element in the first row of the first matrix multiplied by the first element in the first column of the second matrix, then plus ππ because thatβs the second element in the first row of the first matrix multiplied by the second element in the first column of the second matrix.
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And then if we move along to the next element, what we get is ππ plus πβ.
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And thatβs because what weβve got is the first element in the first row of the first matrix multiplied by the first element in the second column of the second matrix plus the second element in the first row of the first matrix multiplied by the second element in the second column of the second matrix.
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And then, if we carry on this pattern, weβll have ππ plus ππ.
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And then for our final element, weβll have ππ plus πβ.
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Okay, great.
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So we remind ourselves how we multiply our matrices.
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So now what weβre going to do is work out what π΄π΅ and π΅π΄ are.
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So weβre gonna start with π΄π΅.
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So itβs the matrix one, two, three, four multiplied by the matrix one, two, one, π.
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So for the first element, weβre gonna have one multiplied by one plus two multiplied by one.
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Then for the next element, weβve got one multiplied by two plus two multiplied by π.
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Then weβll have three multiplied by one plus four multiplied by one.
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And then finally, three multiplied by two plus four multiplied by π.
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So this is gonna give us the matrix three, two plus two π, seven, six plus four π.
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So great, weβve found out what π΄π΅ is.
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So now letβs have a look at π΅π΄.
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And then for π΅π΄, what weβre gonna have is matrix one, two, one, π multiplied by the matrix one, two, three, four.
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And this is gonna be equal to one multiplied by one plus two multiplied by three, one multiplied by two plus two multiplied by four, one multiplied by one plus π multiplied by three, then finally one multiplied by two plus π multiplied by four.
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And what this is gonna give us is the matrix seven, 10, one plus three π, two plus four π.
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Well, we can see straightaway that π΄π΅ cannot be equal to π΅π΄.
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And thatβs because if we have a look at the matrices that weβve got here, our first element in π΄π΅ is three and the first element in π΅π΄ is seven.
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And even if we tried with the other elements, if we take a look at the next element along, weβve got two plus two π and 10.
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Well, if we equated these, we get two plus two π equals 10.
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So two π would be equal to eight.
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And π would be equal to four.
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Then we substituted back in this value for π, weβd get all different values for the different elements.
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So therefore, we can say that in answer to the question, βIs it possible to choose a value for π so that π΄π΅ equals π΅π΄?β we can say that there is no possible choice for π because π΄π΅ cannot be equal to π΅π΄.