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Given that matrix π΄ is negative two, negative nine, eight, seven, find the inverse of the transpose of matrix π΄.
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When trying to solve this problem, we can actually notice two bits of notation here.
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Weβve got π΄ to capital π, and then that is all in our parenthesis to negative one.
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So what I actually have is two things.
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We have the transpose and the inverse.
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And weβre gonna actually have to find both of those to solve this problem.
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We can start first of all with the transpose.
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So when weβre looking to actually transposing a matrix, what we can actually say is for each element of ππ₯π¦ where π₯ is the π₯th row and π¦ is the π¦th column.
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In our matrix π΄, the matrix π΄π, so our transposed matrix, has an equal element at ππ¦π₯, so this time at the π¦th row and the π₯th column.
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So what does that actually mean in practice?
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Well in practice, what it actually means is that weβre going to turn the rows into their corresponding columns.
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For example, weβre gonna to turn the first row of our matrix into the first column of its transpose.
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So if we take a look at our matrix π΄, so what weβve actually got is that matrix π΄ is negative two, negative nine, eight, seven.
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And weβve got our first row, which is negative two, negative nine.
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And weβve actually then turned that into our first column of our transpose matrix.
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So itβs gonna to be negative two, negative nine.
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And now weβve done the same thing with our second row.
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So our second row β eight, seven β has become our second column of our transposed matrix, again eight seven reading downwards.
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So therefore we could say that π΄π is equal to negative two, eight, negative nine, seven.
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So now weβre gonna have a look at the inverse of a matrix cause we wanna see how do we find that.
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Well first of all, a bit like when you multiply a number by its inverse you get one, with matrix if you multiply a matrix by its inverse, you actually get πΌ where πΌ is actually called the identity matrix which can be represented by the matrix one, zero, zero, one.
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Well, okay!
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So we know that thatβs the case, but how weβre actually gonna find the inverse of our matrix.
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Well, actually what weβve got is a two-by- two matrix.
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And thereβs a special formula to help us find the inverse of a two-by-two matrix.
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And what this formula actually tells us is that if we have a matrix in the form π, π, π, π, then the universe of this matrix is equal to one over ππ minus ππ, which is the determinant of that matrix, multiplied by the matrix π, negative π, negative π, π where weβve swapped our π and π values and our π and π elements are now negative.
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Okay, so now weβve got this formula.
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Letβs use it to actually find the inverse of the transposition of matrix π΄.
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Therefore, as our transpose matrix π΄ is going to be negative two, eight, negative nine, seven, then the inverse of that transposition is gonna be equal to one over and then itβs π multiplied by π, which is negative two multiplied by seven, and then π multiplied by π, which will be eight multiplied by negative nine.
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Okay, and then this is all multiplied by the matrix, with our first element as seven because thatβs our π.
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Our second element is going to be a negative eight because itβs the negative of the π element, so negative eight.
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And then our third element, the bottom left element, is gonna be nine because our π element in the transpose matrix is negative nine.
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And this one is negative π, so negative negative nine gives us positive nine.
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And then our final element is negative two.
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Okay, great!
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So letβs calculate.
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So we get that itβs all equal to one over 58 multiplied by the matrix seven, negative eight, nine, negative two, which gives us the matrix seven over 58 negative eight, over 58, nine over 58, negative two over 58.
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And therefore, if we simplify our fractions fully, we can say that given that the matrix π΄ is equal to negative two, negative nine, eight, seven, then the inverse of the transposition of matrix π΄ is gonna be equal to seven over 58, negative four over 29, nine over 58, and negative one over 29.