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 7 20 negative 21. The second row is negative 206. The third row is 50, negative 15. Our answer choices are a negative B 85 C D zero. The first thing to recall is that if we are looking to find the determinant of a three by three matrix, we should think of this as the top row being ABC. The second row being D E F in the third row being G H I. That way we can set this up in a pattern. The determinant comes from the value of a multiplied by the determinant of A two by two matrix that is formed from the rows that A does not belong to the rows in the columns. So in this case E F is the first row, I is the second row in our two by two. This is now subtracted by the value of B multiplied by the determinant of A two by two matrix that B does not belong in. So that would be D F over G I and plus C multiplied by the determinant of another two by two matrix. And again, it doesn't include the rows or columns that C is in. So this would be D E as the top row and G H as the second row. So now that we have this set up, we can plug in the numbers we are given. So starting with a, in our example would be seven and this is going to be multiplied by the determinant of the two by two matrix. That would be 06. And the second row would be zero negative 15. Next we subtract B which in this case is multiplied by the determinant of the two by two matrix. That is D F. So negative 26 is our first row and G I so five negative 15 is our second row plus the value of C which is negative multiplied by the determinant of the two by two matrix from D E. So negative 20 and the second row would be G H. So 50, the next thing to recall is that when you have a two by two matrix, you find the discriminate by doing the product of the top left number and the bottom right number minus the product of the top right number and the bottom left number. So we go diagonally. So we will have seven multiplied by and this will be zero times negative 15, which is zero minus zero, multiplied by six, which is zero, close parentheses. We'll clean that up in the next step minus times in parentheses. Now, negative two multiplied by negative 15 is minus five, multiplied by six is 30 closed parentheses plus a negative 21 multiplied by. And our determinant from this two by two is found by doing negative two multiplied by zero, which is zero minus five, multiplied by zero, which is zero. All right. So let's clean up what's in our parentheses and do our multiplication. Um So we have seven times zero. OK. Minus 20 and 30 minus 30 is again zero plus, actually, I'm gonna rewrite that as minus for my negative. So negative or minus 21 multiplied by zero. So everything's multiplied by zero, which is zero. So I have zero minus zero minus zero, which is zero. So our answer here, the discriminant of this three by three matrix is answer choice D zero. Have a nice day.

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

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