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Find all values of ๐ฅ for which the determinant ๐ฅ, negative two, negative two, ๐ฅ plus the determinant six, three, one, eight equals 45.
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So these lines do not mean absolute value, they represent the determinant, and the way to find the determinant of a matrix of numbers is to take ๐๐ minus ๐๐.
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Essentially, youโre kind of cross multiplying and subtracting, ๐ times ๐ minus ๐ times ๐.
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So letโs go ahead and look at our equation.
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To take the determinant of the first matrix, ๐ฅ times ๐ฅ is ๐ฅ squared minus negative two times negative two which is positive four.
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And now weโre adding the determinant of the other matrix, six times eight, which is 48, minus three times one, which is three.
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And we set it equal to 45.
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So now we need to combine like terms.
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48 minus three is 45.
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Now thereโs two ways we could solve from here.
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Since thereโs 45s in both sides of the equation, we could subtract it.
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They cancel, and we get ๐ฅ squared minus four equal zero.
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And this is a difference of squares because ๐ฅ is being squared and four is a perfect square, itโs two squared.
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So this was factored to be ๐ฅ plus two, ๐ฅ minus two, and we set each factor equal to zero.
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And now we solve.
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So we subtract two from the first equation and add two to the second equation.
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So we get ๐ฅ equals negative two or ๐ฅ equals positive two.
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The other way we couldโve solved from this point is to add negative four and 45.
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And we wouldโve had ๐ฅ squared plus 41 equals 45.
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And now we subtract 41 from both sides of the equation.
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So we get that ๐ฅ squared equals four.
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And now we square root both sides.
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And we get that ๐ฅ equals plus or minus two, which is the exact same thing that we got before.
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So our final answer is: ๐ฅ equals negative two or ๐ฅ equals positive two.