Find the inverse, if it exists, for each matrix. [3x3 matrix]

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Hello, everyone. We are asked to determine the inverse of the following matrix. If possible, we are given a three by three matrix. Recall that to find the inverse of this matrix, we need to find the determinant of it. So we can do one divided by the determinant. And we're gonna multiply that by the adjoint matrix. So first, we're gonna start by finding the determinant. So recall that determinants are created by doing each element of the first row multiplied by the determinant of the minor matrix. That is not including the rows and columns that have that element. Also recall that we have to go positive negative positive. So we alternate positives and negatives. So this will be a negative eight. And then this is uh minor matrix is the first row negative 1, 10 is the second row. And then we have plus a negative multiplied by the minor matrix +84 as the first row negative one negative one as the second row recall that to find the determinant of a two by two matrix, we find the product from left to right and subtract the product from right to left. So we will have eight multiplied by 40 when we do the determinant there and then negative eight multiplied by 80 when we do the determinant of the second one and then negative 12 multiplied by negative four. So cleaning this up, eight, multiplied by 40 is 320 minus eight multiplied by 80 which is 640 plus 48. So our determinant here is negative 272. So that'll be an interesting number to use later. But we're gonna keep that to the side while we work on finding our adjoint matrix to find our adjoint matrix. We first need to find a cofactor matrix and the cofactor matrix will be comprised of minor matrices that will exclude the row and column of the corresponding element. So the first row, first element is eight. So we're gonna exclude the rows and columns that have that element eight. And we get the minor matrix 40 for the first row negative 10. For the second row. For the next element, we're gonna exclude the rows and columns with that second eight. So we have a minor matrix, first row 80, 2nd row negative 1 10. For the third element, we're gonna exclude the rows and columns for the negative 12. And we will have a matrix that is 84 for the first row negative one negative one. For the second row recall that we alternate positives and negatives. So if we start that our first element is positive. I need to put a negative sign in front of my second element. Now I'm ready to move to my second row. So excluding the rows and columns with that eight produces a minor matrix. Eight negative 12 is the first row and negative 1 10. As the second row. Next matrix, we're gonna exclude the rows and columns with the four. So we get eight negative 12 is the first row, negative 1. 10 is the second row. And the third matrix will exclude the rows and columns with the zero. So we have 88 as the first row and negative one negative one is the second rule going back to adding in my negatives. If the last element of the first row is positive, the first element of our second row is negative, then positive, then negative to the third row. We're gonna again create a minor matrix excluding the rows and columns that include this negative one. So we have eight negative 12 as the first row 40 is the second row. Now, excluding the second negative one, we end up with the first row of eight, negative 12 in a second row of 80. And the third matrix there, we're gonna exclude the rows and columns including that 10. We have 88 for the first row, 84 for the second row. And I need to recall uh we ended with a negative in the sec last element of the second row. So that means positive negative. So the second element of the third row is our negative. Now that we have all of that figured out, we are gonna find the determinant of each element for each minor matrix we just wrote. So again, recall you multiply diagonally from left to right and then subtract the product from right to left. And when you do so your top row will be negative 80 negative four. Second row is negative 68 zero. The third row is 48 negative 96 negative 32. Make sure you multiply in those negative signs that we added after. So we can make sure we keep track of our signs. So the last part before we could say we have our adjoint matrix is we need to transpose our rows and columns. So this means our first row. So 40 negative 80 negative four becomes our first column. So it's now a column, 40 negative 80 negative four going down instead of a cross, our second row becomes our second column. So negative 68 zero and our third row becomes our third column again going down, not across. So 48 negative 96 negative 32. So now we have our determinant, we have our adjoint matrix, we can find the inverse matrix. So we are doing one divided by the determinant which was negative 272 multiplied by that ad a joint matrix we just found. So top row 40 negative 68 second row negative 80 68 negative 96 third row negative 40, negative 32. Recall that when you multiply a matrix by a value, a scaler, if you will, we have to multiply each element by that value. So we will get the matrix that has negative 40/272 is the first element negative 68 divided by negative 272. Is your second element in the first row. 48 divided by negative 272 is the third element. Next row, we have negative 80 divided by negative 272. Then 68 divided by negative then negative 96 divided by negative 272. Third row, negative four divided by negative 272. Uh 0 divided by anything is zero and negative 32 divided by negative 272. Obviously, these things can be reduced and so we will be reducing them. All right. So reducing doesn't look like everything gets reduced by the same thing. But when we reduce our first row is negative 34th, next album will be 1/4 and then negative 3/17. Second row is 5/ negative 1/4 6/ third row is 1 68th zero and 2/17. And that is our inverse matrix if we were to multiply that by our original matrix, we should end up with an identity matrix. Have a nice day.

Find the inverse, if it exists, for each matrix. [3x3 matrix]

Key Concepts

Matrix Inversion

Determinant

Row Reduction

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