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Determine if any of the following matrices are inverses for1[3 % Ans: Yes[9 7] Ans: NoAns: NoAns: NoFor each of the following matrices, find the rank: Then. calcula...

Question

Determine if any of the following matrices are inverses for1[3 % Ans: Yes[9 7] Ans: NoAns: NoAns: NoFor each of the following matrices, find the rank: Then. calculate the inverse it exists:A4=(b) B =Ans: rank(A) = 2Ans: rank(B) =Ans: rank(C) = 2 Inverse does not exist _B-1A-1 =[5 :]

Determine if any of the following matrices are inverses for 1 [3 % Ans: Yes [9 7] Ans: No Ans: No Ans: No For each of the following matrices, find the rank: Then. calculate the inverse it exists: A4= (b) B = Ans: rank(A) = 2 Ans: rank(B) = Ans: rank(C) = 2 Inverse does not exist _ B-1 A-1 = [5 :]



Answers

Determine whether each pair of matrices are inverses of each other.
$$
A=\left[\begin{array}{ll}{6} & {2} \\ {5} & {2}\end{array}\right], B=\left[\begin{array}{rr}{1} & {1} \\ {-\frac{5}{2}} & {-3}\end{array}\right]
$$

On this question. We want to determine whether this pair of matrices are in verses or not. So if they are in versus then when we multiply them together, we should get the identity matrix, which looks like this. So this is what we want to get when we multiply them together. And if we d'oh than there in verses. So the best way to multiply major sees if you take, is if you take your 1st 1 and then he right, the 2nd 1 above and to the right of it. And then from here, what happens is that your answer is going to show up in this space right here. So what you want to do is create a little grid. So you're gonna draw straight down between the columns in the matrix on top and between the rows over here. Then it's easier to see where your numbers are coming from. So this number right here is gonna come from this column and this road. So what you dio is you multiply the first in the 1st 2nd and the second and add them. In other words, that would be too times 1/2 plus zero times negative one. So two times 1/2 is one, and then we get zero. So that first element will be one then, for this spot. We're switching to this column, but were still with the first row. So I'm gonna have zero times two plus negative one times negative. 1/3. This goes to zero, and then I have negative one times negative 1/3 which is just positive. 1/3 now for this spot. We're switching to the second row, so we would have 1/2 times one plus negative, three times zero. So 1/2 times one is 1/2. The other part of it is zero, and then zero times one is zero plus negative. Three times negative. 1/3 which is one. We did not get the identity, so we don't have to bother going the other direction. So you want to check and see if a Times B is the same as B times? A. Which is the identity. This is what you're checking. But in the first direction we went, we got this matrix right here, which is not the identity, so we don't need to bother going the other direction. We already know these two are not in Versace

If we have a square matrix, say em with the elements a B c D. The inverse of em, which we denote as M to the negative one, is a matrix such that m times the inverse will give us the identity matrix. So when we're trying to find the inverse of a two by two matrix, the first step we need to do is to calculate something called The Determinant of the Matrix. Determinant is denoted by these bars with the letter of the Matrix inside, and it's defined to be the product of this main diagonal minus the product of this B and C. So let's take a look at how to find the inverse of a couple of different matrices. So let's start with Let's Say, Matrix A, which is 74 five three. So the first thing we want to do is to evaluate the determinant of a So remember that is the product of the main diagonal seven times three minus this product here four times five, and that would give us 21 minus 20 which is one now. The way that we calculate the inverse of a is we multiply a or some iteration of a by the reciprocal of our determinant, which is so this would be 1/1 and what we're multiplying it too, is we take these elements on the main diagonal and we interchange them. So this would be the three up here, and the seven comes down here and then for these other two, we're going to take the negative of them, So this would be a negative four and negative five. So that is just the inverse there. Three negative four negative. Five seven. Let's consider Matrix B, which has the elements 23 45 So first figuring out the determinant which again is the product of the elements in the main diagonal two times five minus the product of these elements three times four. So 10 minus 12 is negative. Two. And when we go to evaluate the inverse of be, what we're going to do is multiply by the reciprocal of our determinant, which was the negative, too. And we're gonna multiply it to the main diagonal. We're going to interchange these. So this would be a five here and a two down here and then these other elements we're going to take the negative, them negative of them. Negative three and negative four. And so when we do this multiplication, we would get negative. Five halfs positive. Three halfs Positive two and negative one. Let's consider Matrix C with elements four Negative six negative. 23 Yeah. Calculating the determinant of C multiplying the elements of our main diagonal four times three and subtracting the product This diagonal negative six times Negative two. So this we get 12 minus 12 and our determinant is zero. Because our determinant zero this means that C does not have an inverse Mhm. Alright, let's try one more. How about matrix D? Yeah. Five negative 26 Negative three. So the determinant of D would be five times negative. Three minus negative two times six. So it gives us negative 15 plus 12, which is negative. Three. So to find the inverse of d, we would multiply by the reciprocal of that determinant. And again we interchange the elements in our matrix D that are on the main diagonal and then these other two elements we take the negative of them. So this would be a positive two in a negative six and then doing this scale or multiplication, we would get one negative two thirds positive too and negative. I think

If matrix A. Is a two by two. Matrix 4354 matrix B. Is 4 -3 -5. positive for we want to find a B. A B will be found By multiplying row one column 1 together. So we get 16 plus negative 15. That's one. And then row one column to that's negative 12 plus 12. That's zero. And then road to column one that's negative 20 plus negative 20. That's negative 40. Um I looked at that incorrectly. That's 20 plus negative 20. That is zero and then negative 15 Plus 16. That is one. Yeah. And then be A. B. A. It would be multiplying Row one Column 1 that's 16 in -15. 01 in Column two That's 12 and -12. And then road to column one. It's negative 20 plus 20 In Row two column To us -15 and positive 16. Mhm. To get the two x two identity matrix in each direction. Yes. B. is the inverse of a. Okay.

Given matrix A is the two by two matrix negative 12 negative 38 in matrix B is negative for one negative 21 half one. To determine that B is the universe of A. By multiplying A times B. So that's rho one times column one. That's four plus negative for which is zero. Which is already tells me that this is not going to be because we're not going to be the identity matrix 1001


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