Video Transcript
Find all values of ๐ฅ for which the determinant ๐ฅ, negative two, negative two, ๐ฅ plus the determinant six, three, one, eight equals 45.
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. Essentially, youโre kind of cross multiplying and subtracting, ๐ times ๐ minus ๐ times ๐. So letโs go ahead and look at our equation.
To take the determinant of the first matrix, ๐ฅ times ๐ฅ is ๐ฅ squared minus negative two times negative two which is positive four. And now weโre adding the determinant of the other matrix, six times eight, which is 48, minus three times one, which is three. And we set it equal to 45. So now we need to combine like terms. 48 minus three is 45. Now thereโs two ways we could solve from here. Since thereโs 45s in both sides of the equation, we could subtract it. They cancel, and we get ๐ฅ squared minus four equal zero. And this is a difference of squares because ๐ฅ is being squared and four is a perfect square, itโs two squared. So this was factored to be ๐ฅ plus two, ๐ฅ minus two, and we set each factor equal to zero. And now we solve.
So we subtract two from the first equation and add two to the second equation. So we get ๐ฅ equals negative two or ๐ฅ equals positive two. The other way we couldโve solved from this point is to add negative four and 45. And we wouldโve had ๐ฅ squared plus 41 equals 45. And now we subtract 41 from both sides of the equation. So we get that ๐ฅ squared equals four. And now we square root both sides. And we get that ๐ฅ equals plus or minus two, which is the exact same thing that we got before.
So our final answer is: ๐ฅ equals negative two or ๐ฅ equals positive two.