Find the cofactor of each element in the second row of each matrix. See Example 2.

Question

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Hello, everyone. We are asked for the following matrix evaluate the co factors for each element of the second row. The matrix we are given is a three by three. The first row reads 34, negative three. The second row reads 45, negative four. The third row reads negative 383. Our answer choices are a negative 0, 36 B 0, negative 36 C 0, 36 D negative 0, negative 36. First, I would like you to recall that when we were working with matrices, we have a format that goes plus minus plus for the first row minus plus minus for the second row and plus minus plus for the third row when we are trying to find the co factors. So this means if I want the co factor of four, which is in the first column, second row, I'm going to start by creating a matrix that will be comprised of the rows and columns that do not include that four. So if I want the co factor of four, I know this is going to be a negative because that's its place in our general format. And then I'm gonna make a matrix of the rows and columns that are not included. So we would have four negative three and the second row would be from there. I can find the determinant. The negative sign is gonna stay out front. I'm gonna open a set of parentheses. The determinant of a two by two matrix is found by subtracting the product of the diagonals starting in the top left corner. So the product of four and three is minus the product of negative three and eight, which is negative 24. I put the negative 24 in parentheses just to separate it from my, my sign solving this, I have a negative sign and then in my parentheses 12 minus negative 24 which would be a positive 36. So our minus that's on the outside or negative on the outside of the parentheses times 36 means the cofactor of the value four in the second row is negative 36. We're gonna repeat this process to find the co factor of the value five from the second row. That would be in a positive location from our general matrix. So we do not need to put a minus sign out front. And again, this will be comprised of the rows and columns that do not include the value five. So in this case, the first row would be three negative three and the second row would be negative 33. Again, we're gonna find our determinant by subtracting the diagonal products. Three, multiplied by three is nine minus the other diagonal, which would be negative three multiplied by negative three, which is a positive nine, nine minus nine is zero. So the cofactor of five in the second row is zero. Next, we need the co factor of the value negative four from the second row. And again, looking at our generalized format, this should be a negative value. So I'm gonna put a negative in front of my matrix. And this again will be comprised of the rows and columns, not including that negative four. So our first row is 34, our second row is negative 38. I'm going to put the minus sign or the negative sign out front, open a set of parentheses and find my determinant by doing the diagonals. So three multiplied by eight is minus four, multiplied by negative three, which is negative 12. I put the negative 12 in a set of parentheses just to separate it from our minus sign simplifying in the parentheses. I still have a negative sign out front 24 minus negative 12 is like adding. So that will be a positive 36. And when I bring the negative sign in, this will be a negative 36. So our co factor for the negative four that is in the second row is negative 36. So we now have all three co factors. We have negative 36 0, negative 36. This matches answer choice. D have a nice day.

Find the cofactor of each element in the second row of each matrix. See Example 2.

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Key Concepts

Cofactor

Matrix

Determinant