Use the determinant theorems to evaluate each determinant. See Ex...

Question

Use the determinant theorems to evaluate each determinant. See Example 4.

Accepted Answer

Hello, everyone. We are asked to find the value of the given determinant. Using the determinant theorems. We have a three by three matrix. The first row is negative 626. 2nd row is 301, 3rd row is 036. Our answer choices are a negative B zero C D six. First thing to remember is how to find the determinant of a three by three matrix. So if we have a three by three matrix, and we want to think of this as the top row being ABC, the second row being D E F in the third row being G H I. Our formula for this is the value of a multiplied by the determinant of E F H I as a two by two matrix minus the value of B multiplied by the determinant of another two by two matrix which would be D F G I plus the value of C multiplied by the determinant of a third two by two matrix which would be D E G H. For me, it was easier to remember which values went in the two by two matrix as they are the values that are not in the same row or column as the number, I'm multiplying it by. So let's set this up looking at our original matrix A is going to be negative six. So I'm going to treat this as our determinant is gonna be equal to negative six multiplied by. And then we need the determinant of the matrix that is from E F. So 01 for the top row and H I, which would be minus the value of B which is two multiplied by the determinant of the two by two matrix D F. So D F would be 31, their second row is G I, so plus the value of C which in our matrix is six multiplied by the determinant of the two by two matrix. Uh D E in our case is 30. And then the second row is G H which is 03. The next thing you need to recall is that when you have a two by two matrix, the determinant is found by diagonally multiplying the values together. So starting at the top left, top left multiplied by the bottom, right minus the top right, multiplied by the bottom left. So in our case, we have negative sex and then I'm gonna put in parentheses. So I'm multiplying by zero, multiplied by six, which is zero minus one, multiplied by three, which is three closed parentheses minus two times parentheses. The determinant of our second matrix. So that would be three multiplied by six, which is 18 minus zero, multiplied by one, which is zero, close parentheses plus six, multiplied by another set of parentheses. And in this one, the determinant of that third two by two matrix. So three multiplied by three is nine minus zero. Multiplied by zero is zero, close parentheses. Now I'm gonna combine what I can within the parentheses. So then I can do my multiplication. So I have negative six multiplied by negative three minus two, multiplied by 18 plus six multiplied by nine. So now I'm gonna do out the multiplication negative six multiplied by negative three is a positive 18, negative two multiplied by 18 is negative and then six multiplied by nine is 54. And when I add that all together, I am left with 36. So the value of the determinant of this three by three matrix is 36 which is answer choice C have a nice day.

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Use the determinant theorems to evaluate each determinant. See Example 4.