Video Transcript
Find the solution set of the determinant of negative eight 𝑥, seven 𝑥, three 𝑥, negative two 𝑥, zero, negative five 𝑥, negative eight 𝑥, nine 𝑥, three 𝑥 equals 736 in the set of real numbers.
In order to be able to solve this matrix equation, we’re going to begin by finding the determinant of this three-by-three matrix. So let’s remind ourselves how to do that. Given a three-by-three matrix 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, its determinant is 𝑎 times the determinant of the two-by-two matrix 𝑒, 𝑓, ℎ, 𝑖 minus 𝑏 times the determinant of the two-by-two matrix 𝑑, 𝑓, 𝑔, 𝑖 plus 𝑐 times the determinant of 𝑑, 𝑒, 𝑔, ℎ. Essentially, we take each element on our first row and we multiply it by the determinant of the two-by-two matrix that remains when we remove that row and that column. And for our second element 𝑏, we subtract 𝑏 times that two-by-two determinant. So with this in mind, let’s find the determinant of our three-by-three matrix.
We begin with multiplying negative eight 𝑥 by the determinant of the two-by-two matrix zero, negative five 𝑥, nine 𝑥, three 𝑥. Then, we’ll subtract seven 𝑥 and we’ll multiply that by the determinant of the two-by-two matrix whose elements are in the first and third column. Finally, let’s add three 𝑥 times the determinant of the two-by-two matrix negative two 𝑥, zero, negative eight 𝑥, nine 𝑥. So we now need to work out the determinant of these two-by-two matrices. Well, the determinant of a matrix 𝑎, 𝑏, 𝑐, 𝑑 is the product of the top-left and bottom-right element minus the product of the top right and bottom left.
So for our first determinant, we’re going to calculate zero times three 𝑥. And then we’re going to subtract negative five 𝑥 times nine 𝑥. Well, zero times three 𝑥 is zero. So it’s negative negative five 𝑥 times nine 𝑥, which is 45𝑥 squared. Then, the determinant of the remaining two matrices are negative 46𝑥 squared and negative 18𝑥 squared. So the determinant of our three-by-three matrix is negative eight 𝑥 times 45𝑥 squared minus seven 𝑥 times negative 46𝑥 squared plus three 𝑥 times negative 18𝑥 squared, which is equal to negative 92𝑥 cubed. And so we can substitute this determinant back into our original equation, and we get negative 92𝑥 cubed equals 736.
Dividing both sides of this equation by negative 92, and we get 𝑥 cubed is equal to negative eight. So 𝑥 is equal to the cube root of negative eight, which is simply negative two. Since the solution to our matrix equation is 𝑥 equals negative two, we can say that the solution set is the set containing the single element negative two.
