Find the cofactor of each element in the second row of each matri...

Question

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

Accepted Answer

Hello everyone. For the following matrix, we are asked to evaluate the co factors for each element of the second row, we're given a three by three matrix. The first row of that matrix reads negative 705. The second row is 370. The third row is 873. Our answer choices are a 61 49 B negative 35 negative 61 C 35 negative 61 negative 49 D 35 negative 61 49. Before we start solving this recall that when we were working with the cofactors of a matrix, we have a sign chart that we need to follow in that sign chart. The first row is a positive negative positive. The second row is negative positive negative. The third row is positive, negative positive. So in this example, we are asked about the elements of the second row. So we need to follow the sign chart that matches the second row. So the first element in the second row is the value three. So we need to find the co factor of three in the second row. First from my sign chart, I know this needs to be a negative. So I'm placing my negative sign next to find a cofactor. We need to find the minor matrix that is comprised of the rows and columns that do not include the element three from the second row. So in this case, the first row in our minor matrix would be the value 05. And our second row would be the value 73. From here. We want to find the determinant of this two by two. Matrix recall we do that by finding the difference of the products of the diagonals starting in the top left corner. So my negative sign isn't changing. So I have a negative sign and then open parentheses and then multiplying diagonally starting in the top left zero, multiplied by three is zero minus five, multiplied by seven, which is 35. So we now have a negative sign multiplied by the parentheses, zero minus 35. Simplifying in my parentheses, I still have my negative sign out front zero minus 35 is negative 35. And when I do a negative multiplied by negative 35 I get that the co factor of the element three in the second row is 35. Next, I need to find the co factor of the element seven from the second row. This one has a positive on our sign chart. So I do not need to put a negative sign in it. My minor matrix again, will be comprised of the rows and columns that do not include that element seven. So my top row is negative 75 and my bottom row is 83. And again, I'm going to multiply diagonally. So negative seven multiplied by three is negative 21. And I'm going to subtract the other diagonal product five multiplied by eight is 40. So we have negative 21 minus 40 which is negative 61. So that is my co factor for the element seven in the second row. One more, we need to find the co factor of zero from that second row. First, I look at my sign chart and I know this needs to be a negative. So I put my negative before my minor matrix and my minor matrix will be the values again from the rows and columns that do not include that element. So my top row is negative 70. My bottom row is and now I'll find that determinant. I'm gonna keep my negative sign out front open a set of parentheses negative seven multiplied by positive seven is a negative minus my other diagonal product. So zero multiplied by eight is zero. Simplifying in my parentheses, I still have my negative sign out front, negative 49 minus zero is still negative and a negative multiplied by negative 49 is positive 49. So I now have all three of my cofactors I have negative, 61 49 and that matches answer choice. D have a nice day.

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