Evaluate each determinant. See Example 3.

Question

Accepted Answer

Hello, everyone. We are going to find the value of the determinant given below. We are given a three by three matrix across the first row. We have the values 26 3. The second row is negative 3 12 7. The third row is negative 7 24 20. Our answer choices are a 8850 B negative 8850 C 1542 D negative 1542. First thing I'm gonna do is I'm gonna write a three by three matrix with general values. So my first row is ABC second row D E F third row G H I, this way I know which value I am referring to in my format formula. So the formula define the determinant of the three by three matrix. I have the value of a multiplied by the determinant of the minor matrix which is a two by two. And that's gonna be comprised of the rows and columns that do not include A. So in this case, that would be the first row is E F second row is H I then minus the value of B multiplied by the determinant of the minor matrix that will be comprised of the rows and columns that do not include B. So this will be the first row is D F. The second row is G I and then plus the value of C multiplied by the determinant of the two by two matrix again comprised of the values from the rows and columns that do not include C. So here, this should be the first row is D E second row is G H. Now that we have this, we're gonna fill it in with our values from the matrix we're given. So A would be 20 multiplied by the determinant of the matrix. That would be E F so 12 7 for the first row H I so 24 20 for the second row minus our B value is six multiplied by the determinant of the two by two matrix. That would be D F. So negative 37 for our first row and G I so negative 7 20 for our second row plus the value of C which is three multiplied by the determinant of the matrix D E. So negative 3 12 for the first row and G H which is negative 7 for our second row. Next recall that we need to find the determinant of A two by two matrix, we find that by multiplying diagonally starting in the top left corner and then subtracting the diagonal from the top right corner. So in this case, this would look like 20 multiplied by, we'd have 12 multiplied by 20 which is 24. Oh my gosh, 24 240. Where's my brain minus seven multiplied by 24? And that would be 168 closed parentheses minus six, which will be multiplied by again, the determinant which would be negative three multiplied by 20 which is negative 60 minus seven multiplied by negative seven, which is negative 49. I put the negative 49 in parentheses just to separate it from our minus sign plus three should be multiplied by the product of negative three and 24 which is negative 72 minus the product of 12 and negative seven, which is negative 84. Again, I put negative 84 in parentheses to separate it from my minus sign. At this point, we can do all of the subtraction that is inside the parentheses and then we'll do our final multiplication. So we have 20 multiplied by and we have 240 minus which is 72 minus six which will be multiplied by. In parentheses, we have negative 60 minus a negative 49. So double negatives are like adding. So I'm gonna treat this as negative 60 plus 49 which is negative 11 closed parentheses plus three multiplied by and we have negative 72 minus negative 84. Again, double negatives are like adding. So I'm treating this as negative 72 plus 84 which is a positive 12. So now we are going to finish multiplying these together and then we'll add them up. So we have 20 multiplied by 72. So that should be and then negative six multiplied by negative 11 would be plus 66 plus three multiplied by 12, which is 36. And when I add all of those values together, it looks like I end up with the value 1542. That is the determinant of this matrix. And it's answer choice C have a nice day.

Evaluate each determinant. See Example 3.

Key Concepts

Determinants

Matrix Operations

Cofactor Expansion

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