WEBVTT
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Find numbers π, π, and π so that π multiplied by the matrix one, one, zero, negative one.
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Plus π multiplied by the matrix one, zero, zero, one.
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Plus π multiplied by the matrix zero, negative one, one, zero.
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Is equal to the matrix one, zero, negative one, three.
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So the first thing Iβm gonna do is multiply matrices by either π, π, or π on the left-hand side of the equation.
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Now for scalar multiplication, what we do is we multiply each of our components of our matrix by the scalar value we have.
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So thatβs π.
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So weβre gonna have the matrix π, π, zero, negative π for our first matrix.
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Then for our second matrix, weβre gonna have π, zero, zero, π.
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Thatβs cause we have π multiplied by one, π multiplied by zero, π multiplied by zero, and π multiplied by one.
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And then for our final matrix, weβre gonna have zero, negative π, π, zero.
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So now to find out what the numbers π, π, and π are, what weβre gonna do is weβre going to set up some equations.
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And we can do that by equating the components of our matrices.
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Cause we know, for instance, that π plus π plus zero is gonna be equal to one.
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And thatβs because they are corresponding components in our matrices.
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So our first equation, like we said, is π plus π equals one.
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So now if we take a look at our next equation, weβve got π plus zero plus negative π, well gonna be equal to zero.
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So therefore, weβd have π minus π is equal to zero.
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So then weβre gonna have zero plus zero plus π is equal to negative one.
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So therefore, weβve got that π is equal to negative one.
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And then, finally, what weβve got is negative π plus π is equal to three.
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Okay, great.
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So thatβs our equations.
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So letβs use them to find out π, π, and π.
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Well, we already know that π is equal to negative one.
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So now what weβre going to find next?
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Well, if we have π minus π is equal to zero, and then we add π to each side of our equation, weβre gonna get π is equal to π.
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So therefore, if we know that π is equal to π, then π is gonna also be negative one.
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So now all we need to do is find out π.
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Well, if we utilised the first equation, we couldβve done the first or the fourth.
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But weβre gonna go for the first.
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Then what weβre gonna get is negative one β and thatβs cause π is equal to negative one β plus π is equal to one.
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So then if we add one to each side of the equation, weβre gonna get π is equal to two.
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So therefore, we solved the problem cause we found the three numbers π, π, and π.
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And they are π is equal to negative one, π is equal to two, and π is equal to negative one.