WEBVTT
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Find the multiplicative inverse of 69, zero, zero, 69.
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Remember, for a two-by-two matrix ๐, ๐, ๐, ๐, the inverse of ๐ด is one over the determinant of ๐ด multiplied by ๐, negative ๐, negative ๐, ๐, where the determinant of ๐ด is found by multiplying ๐ by ๐ and subtracting ๐ multiplied by ๐.
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Notice that this means if the determinant of the matrix is zero, then thereโs no multiplicative inverse, since one over the determinant of ๐ด will be one over zero, which we know to be undefined.
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Letโs begin then by checking that there is an inverse for this matrix and finding its determinant.
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We start by multiplying the element on the top left by the element on the bottom right.
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We then subtract the product of the elements on the top right and the bottom left.
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And our determinant is 69 multiplied by 69 minus zero multiplied by zero, which is 4761.
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Since the determinant of our matrix is not equal to zero, then we know the multiplicative inverse does indeed exist.
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Letโs substitute what we know about our matrix into the formula for the inverse.
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Itโs one over 4761 multiplied by 69, zero, zero, 69.
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Now you may be thinking that weโve made a mistake because this looks really similar to the original matrix.
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However, if you recall, we switch the top left and bottom right element.
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Since theyโre both 69, this actually doesnโt look like thereโs a difference.
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And then we change the sign of the elements on the top right and bottom left.
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But since theyโre just zero, negative zero is still zero.
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Finally, what weโll do is multiply each element in this matrix by one over 4761.
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And doing so, we get one 69th, zero, zero, and one 69th.
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The multiplicative inverse of 69, zero, zero, 69 is one 69th, zero, zero, one 69th.