In Exercises 13 - 18, use the fact that if
a b d - b
A = then A^(...

Question

In Exercises 13 - 18, use the fact that if a b d - b A = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c a each matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 3 - 1 A = - 4 2

Accepted Answer

hey everyone in this problem. We are asked to find the multiplication inverse of the matrix that were given by using the formula for a inverse. And we want to verify that a inverse is equal to I to an a inverse A is equal to I two. Where I two is a multiplication identity matrix of order to Now the matrix were given A is a two by two matrix first column two negative five. And second column negative 36. Okay. Now we have this equation for a inverse here. Okay, one over A D minus Bc times the matrix with first column D negative C. Second column negative B. A. Okay, so let's first identify what this A B C and D is in our matrix and that's given by the position in the matrix. Ok. So A is going to be equal to two. B is equal to negative three. C is equal to negative five and D is equal to six from our matrix A. Ok. Based on their position in that matrix A. Is the top left. B Is the top right to bottom left. Is the bottom right? Okay, so this means that a inverse can be written as one over a times D So to times six minus B. Negative three times C negative five times the matrix D. Six negative seed five. Okay, that's the first column and the second column has negative B which is going to be three and a which is two. Okay, so let's just go ahead and simplify. We are going to get that. This is going to be we have 1/12 minus 15. So we get one over negative three times are matrix 65 in the first column 32 in the second column. Okay, and let's go ahead because we're gonna be multiplying a inverse and a inverse. A Let's go ahead and take this constant of negative one third into our matrix. And we're going to have negative two negative one along the first row, negative five thirds and negative two thirds along the second row. Okay, Alright, great. So now we want to do the multiplication a inverse and inverse A to verify that they're equal to the identity matrix, which means that this is indeed the multiplication inverse of a. Okay, alright, so let's start with a a inverse. Okay, now a inverse. Well this is going to be equal to Our Matrix. A negative positive two negative five. In the first column negative 36 In the second column K times the inverse we just found Which is negative, 2 -5/ -1 -2/3. Let me just highlight this. So we don't lose track of our inverse matrix when we're using it. Okay, so when we're doing matrix multiplication, what we want to do is take a row of the first matrix and multiply it term by term to a column in the second matrix. Okay, and put that result in the corresponding row and column of our resulting matrix. So what does that actually look like? So let's start with the first row of a in the first column of a inverse. Our resulting matrix, it's going to be the multiplication of these two. We're going to start with the first element in the row two. We're going to multiply by the first element in the column -2. Okay then we're gonna add we're gonna work across the row and down the column. We're going to multiply by the next element in the row negative three times the next element in the column negative five thirds. Okay. And because we've used row one and column one, we're putting this in row one, column one of our resulting matrix. All right, we can work across. So using the same row. Now moving on to the second column, You'll get the first element of the row two times the first element of the column, negative one. Okay? And then we add the next element in the row negative three. Is the next element in the column negative two thirds. Okay, now we're done with the first row, we can move on to the second row. So we're gonna take the second row, go back to the first column and again, first thing in the row negative five times first thing in the column negative two. And we're gonna add next thing in the row six. Next thing in the column negative five thirds. Okay. Alright, so now we just have one term left. That's going to be the second row. Second column we get negative five times negative one plus six times negative two thirds. Okay. And now the hard part is done. We're done The actual multiplication. Now we're just it's just a matter of arithmetic to work out this matrix. Okay. And what we'll see is we get negative for plus five in the first row. First column, that's gonna be one. Okay? First column, second row we have um negative five times negative two gives us 10. Okay? And then we're adding negative 10. So 10 minus 10 gives us zero 1st row. 2nd column we have negative two plus two. Zero 2nd row. 2nd call. We have five -4. This gives us one. And this is the multiplication identity matrix and two dimensions. Okay. Which is what we wanted. So we verified that a inverse does in fact equal I two. So that gets a checkmark or smiley face, whatever you want. Yes, that works out. Okay. Now we have to do the same with a inverse. A Okay, so let's give ourselves a little bit more space. We're going to do the same with a inverse. A Now this is going to equal to a inverse. I'll remind you. First column negative to negative five thirds and second column negative one negative two thirds. We have second column or sorry, second matrix A first column two negative five, second column negative 36. Alrighty. Now we're gonna work this out the exact same way we did for a inverse A or sorry, a universe. We're gonna start with the first row. First column. The first element in the row negative two times the first element in the column to. Okay. And then we add the next element in the row negative one times the next element in the column. Negative five. Okay? We stick with the same row, moving to the other column. First thing in the row negative two times the first thing in the next column negative three. Okay. Next thing in the row -1. Next thing in the column six. Moving on to row two. First thing in row two negative five thirds times the first thing in column 12, we add the next thing in the row negative two thirds. And the next thing in the column negative five. Okay. Moving on to the final row, Final column, we get negative five thirds times three plus negative two thirds times six. Okay. And again, hard parts done. We're done the multiplication. Now. Now we just need to do a bit of arithmetic to work this out. And what we'll see is that we get negative four plus 51 negative 13th plus 13th zero, 6 -60, negative five. I'm sorry. We missed a negative here on this. three. This should be negative three. Okay, so we get positive five minus four. And that's one. There we go. Sorry, mix up that sign. always double check signs are easy to lose track of when you're doing this. Okay? All right. And we'll see that. This gives us the identity matrix in two dimensions. Again. The multiple identity matrix. Okay. And so let's check Mark. Happy Face. Whatever you want to do, A inverse A is also equal to the identity matrix. And so this is in fact the inverse of A that we were looking for. And if we go back up here to answer this question, okay, The inverse A. That we found is going to be B negative two negative five thirds in the first column. Negative one to negative two thirds in the second column. Thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 13 - 18, use the fact that if a b d - b A = then A^(-1) = 1/(ad-bc) to find the inverse of c d - c a each matrix, if possible. Check that AA^(-1) = I_2 and A^(-1)A = I_2. 3 - 1 A = - 4 2