In Exercises 1 - 12, find the products AB and BA to determine whe...

Question

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 0 0 - 2 1 1 2 0 3- 1 0 1 1 0 1 1 1A = B = 0 1 - 1 0 0 1 0 11 0 0 - 1 1 2 0 2

Matrices A and B for Exercise 11 in college algebra, chapter on systems of equations.

Accepted Answer

Hey everyone. Welcome back in this problem. We are asked to work out the products A B and B. A. For our given matrices in order to show if B is the multiplication inverse of A. Okay, so the matrices, A and B were given our four by four matrices. So let's go ahead and start with the first product A B a Beacon. Now this is going to be A. Which has first column 0 to 0, negative two second column 00, negative 20, 3rd column four negative 2 to 0. And final column negative two negative 202. Ok. Times B. Which is again a four by four matrix with first column negative 0. 00 negative 0.5 2nd column negative one, negative 0.5 negative 0. negative one, third column zero negative 0.500. And final column negative 1.5 negative 0.5 negative 0.5 and negative one. All right. So when we're doing multiplication between two matrices, what we wanna do is we want to take the first row of a, multiply it by the first column of B, term by term. And that's going to go in row one column one of our matrix. Okay, then we're going to do the same thing. We're a one of a with column two of B. That's gonna go in row one, column two of our resulting matrix. Okay, so let's see this in action. And it's important to note these are both four by four matrices. Okay, so we're doing matrix multiplication. We need the number of columns of the first matrix to be equal to the number of rows of the second matrix. Ok, So when we write the dimensions, we need the second dimension of a to equal the first dimension of B. Okay. That allows us to do that multiplication. And when we're talking about the inverse, which we're trying to find, we would need these to be exactly the same dimension in order for me to be an inverse. And so they are both 4x4. So we're good there. Alright, so for our multiplication again, we're starting with the first row of a. Okay, so this first row of a times the first column of B. So what we're going to get is the first term times the first term of each. So we get zero from a -0.5 From Me. Okay then we get zero. Again we're gonna add zero time zero. Okay then we're gonna add the next room four Times III term in this column, zero. And then we're gonna add the fourth term in the row to the fourth term in the column, multiplied together. Okay. Alright. And so this is going to be that entry there. Hm. Now we can do the same thing with the next. Yeah. So or sorry, with the next column. So we're gonna take the same row. Okay. But now we're gonna multiply it by the second call. What we're gonna have we're gonna have zero times negative one. That's going to be zero. And I'm not going to write that term just for the sake of space. I'm not gonna write that term here, but we're gonna have zero times negative one. Then we're gonna have zero times negative 0.5. That's also going to be zero. Then we're gonna have four Times -0.5. Okay? Plus negative two times negative one. Alright? We can keep going across the columns. We're gonna have zero times zero, zero times negative 0.5 four times zero and negative two times zero. So in the third column we're going to just end up with zero first row. Okay? Now, for the final column with the first row, zero times negative 1.5 is gonna give us 00 times negative 0.5 is going to give us 04 times negative 0.5. That's gonna give us negative two and then we have negative two minus sorry, times negative one. That's going to be plus two. Okay? So this is also going to give us zero. All right. So we haven't written all the steps just to save space there on those last ones. Let's write out the steps again just to make sure that we know what we're doing. Okay? So here we were going to start with the second row and the first call. We're gonna do this in green. So we're going to start with the first element in the row, we're looking at two. In terms of first element in the column, we're looking at negative 0.5. Okay. Then we're going to go to the second element in the road, we're looking at multiply it by the second element in the column. We're looking at zero. The third element in the row we're looking at is negative two. And the third element in the column we're looking at is zero. And finally, the last element of the row we're looking at is negative two and the last element of the column is negative 0.5. Okay. And so that Is what we get from doing row two times column one. Alright, doing the same thing for column to we're going to get negative two. Okay, so two times negative one is going to be negative 20 times negative 0.5 is gonna be zero, negative two times negative 0.5, that's gonna be one. So we're gonna plus one And then we're going to have negative two Times -1. That's going to be plus two. Alright, third column again, we're going to have zero for each of these entries, so zero all the way around, we're gonna get zero in that column And for the final one we're gonna get two times negative 1.5. So that's gonna be negative three. We're gonna get zero, we're gonna negative two times negative 0.5. That's gonna be plus one. Let me just get rid of this so we can put it a little bit further out. And in the final term the final element of the rho times the final element of the column. We're gonna get negative two times negative one. Which is gonna give us plus two. Right? All right, let's move down and keep on going. We're gonna need it bigger braces for our matrix. Okay? So now we're going to go to the third row. Okay? We're in the third row of our resulting matrix which means we're looking at the third row of our first matrix and we're gonna go column by column. So we get zero times negative 0. Plus -2 times zero Plus two times 0 Plus zero times negative 0.5 In the second column we get zero times negative one. We get negative two times negative 0.5 which is gonna be one, two times negative 0.5 which will be minus one and then zero times negative 10 In the third column we get zero negative two times negative 0.5. That's gonna be one. And then 00 In the final column we get 0 -2 times negative 0.5. That's gonna be one, Two times negative 0.5. That's gonna be negative one and then we have zero times negative. Okay? All right, the last row again, we'll write out that first column directly, completely negative two times negative 0. plus zero times zero, zero time, zero Plus two times negative 0.5. In the second column we get negative two times negative one. That's gonna give us two. Okay? We have zero times negative 0.5. We have zero times negative 0.5 again and then we have two times negative one. That's going to be negative. In the third column negative two times zero is 00 times negative 0.500 times zero and two times zero. So all of these are going to be zero And in the final column negative two times negative 1.5. This is going to give us positive three zero times negative 0.500 times negative 0.50 again and two times negative one is minus two. Okay, so we've done the hard part. We've done the multiplication. We've put everything where it needs to go. And now it's just a matter of adding and subtracting all of these terms within this matrix. Ok, so working this out a little more space, we're going to get the following we have negative two times negative 0.5 is gonna give us one. Okay working down the column we get negative positive one. Okay, so zero, Same thing here, we have zeros in all of these terms. So we get zero and in this final column of row or sorry, in the final row of column one we have 100 and negative one. So one minus one is going to give us zero. Alright, next call. four times negative 0.5 is going to give us - Plus two. Well that's going to be easier. Negative two plus one plus two. Well that's going to be one. One minus 102 minus 20. Okay. Next column, Well we already have this 10010. And in the final column we have a zero negative three plus one plus two. That zero, 1 -1, 0 and 3 -2. 1. Okay. And what you'll notice is that this is the identity matrix? Okay. In four dimensions. Alright, so we found the product A. B. Now we need to do the same for B. A. Okay, so let's give ourselves the full space here and we are going to find the product B. Yeah. And in this case four x 4 Matrix again. And we're doing the same process. We've just lipped around the matrices. So we're on the third row of B. Which is zero negative 0.500. In the final row. Or sorry? Final column negative 1.5 negative 0.5 negative 0.5 negative one times a first column, 0 to 0 negative two zero zero, four negative two -2 -20. Alright, When we did a B we wrote out the first column kind of fully and then when we wrote out the 2nd, 3rd and 4th column, we just did it more straightforward without writing every single term. Okay. Just to make it shorter. So we would fit on the page in this one. Let's do the same. But let's do it with the second row instead. Or sorry, the second column of B. A. Just so you can see it for both columns. Alright, so we get our big resulting matrix here we are going to have negative 0.5 times zero. That's gonna be zero negative one times two. That's gonna be negative 20 times negative two. Or sorry, zero times 00 and negative 1.5 times negative two. That's gonna be plus three. Second row. First column, zero times zero is zero, negative 0.5 times two. That's going to be negative one. -0.5 times zero is going to be zero and negative 0.5 times negative two. That's going to be plus one. Okay. 3rd row, first column, zero times 0 -0.5 times two is negative one. Zero times zero is zero negative 0.5 times negative two is plus one. And the final row in the first column negative, 0.5 times zero is zero negative one times two is negative two. Zero times zero is zero negative one times negative two is plus two. Okay so we've done the first column. Now let's go ahead and do the second column and we're going to write out the details. Let's start with the first row. Second column. Mhm. We're gonna have negative 0.5 time zero plus negative one times zero. Okay. We're working across the row at the same time that we worked down the column plus zero times negative two Plus negative 1.5 times zero. Okay, working across the road down the column, multiplying the corresponding elements and then adding in between. Mhm. Alright let's do the same thing. But for a road to so we're doing the next row down, we're staying in the same column. We're going to get the following first element of the row, zero times the first element of the column zero. Okay now we're adding second element of the ro negative 0. times second element of the column zero. Now we add next third element of the row negative 0.5 times third element of the column, negative two. And finally, fourth element of the rho times fourth element of the call negative 0.5 times zero. And we write it here in this blue circle. Okay? Alright, so that's how we work it out. Let's keep going zero times zero zero, negative 0.5 times 000 times negative 20 negative 0.5 times 00. Okay so that's third row. Second column. Now we're on to the fourth row. Second column doing the same, working across. And working down negative 0.5 times zero is zero, negative one times zero is 00 times negative two is zero negative one times zero is zero. Okay, On to the next columns, -0.5 times four is negative two negative one times negative two is plus two. Zero times two is zero, negative 1.5 times zero is zero. Okay, next row, same column, zero times 40 -0.5 times two is one -0.5 times two is negative one -0.5 times zero is 0. Same column. Working in the third row, zero times 40 negative 0.5 times negative two is positive one, zero times 20 negative 0.5 times zero is zero. Okay? And working in the fourth row, third column negative 0.5 times four is negative two negative one times negative two is two, zero times two is zero and negative one times zero is zero. Alright, we're onto the final column. Starting with the first row negative 0.5 times negative two gives us one negative one times negative two gives us times 00 negative 1.5 times two gives us negative three. Second row, yep. Second row, zero times negative 20 negative 0.5 times negative to positive one negative 0.5 times 00 negative 0.5 times two minus one third row. Last column we're working our way down. Zero times negative two is zero negative 0.5 times negative two is 10 times zero is zero. And negative 0.5 times two is minus one. Okay, last roll last column negative 0.5 times negative two is one negative one times negative two is two, zero times zero is zero and negative one times two is negative two. Alrighty. So there we have it our matrix. And if we simplify we get the following negative two plus three is one minus one plus 10 minus one plus 10 minus two plus 20. Same thing here we're gonna get 00000. And every single one of those terms that zero here we have 0010. So we're gonna have one. Okay negative two plus 201 minus 10. We have a one here and then A zero. Okay. And the final column, one plus two minus three. That's 01 minus one is zero. Again, One minus 101 plus two minus two gives us one. Okay. And so what we get here is also the identity matrix for four dimension matrix. Okay now we found A B. And we found B. A. Let's go back up. Okay we were trying to find whether or not be is a multiplication inverse of A. And what we found is that a. B. Is equal to B. A. And they are both equal to the identity matrix, the fourth dimension. What this means is that B is the multiplication inverse of A. Okay. I hope this video helped. Thanks everyone for watching See you in the next video.

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 0 0 - 2 1 1 2 0 3- 1 0 1 1 0 1 1 1A = B = 0 1 - 1 0 0 1 0 11 0 0 - 1 1 2 0 2

Key Concepts

Matrix Multiplication

Multiplicative Inverse of a Matrix

Identity Matrix

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