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 1 0 0 0 1 A = 0 0 1 B = 1 0 0 1 0 0 0 1 0

Accepted Answer

Hey everyone, welcome back in this problem. We are given two matrices. A and B. Were asked to work with the products A. B and B. A. In order to show if B is a multiple of inverse of A. Okay, So in this case we have four x 4 matrices. Okay, so the dimensions of both the columns and the rows are the same. So there's no issue over multiplying and we know that B could potentially be the multiplication inverse of A. Because they do have the same dimension. Okay, so let's go ahead and get started with A. B. So a is going to be the matrix with first column 1000 second column 0010, 3rd column 0001 and fourth column 0100 M. B is going to be the matrix with first column 1000, 2nd column 0001, third column 0100 and fourth column 0010. Okay, when you're multiplying matrices like these that are a little bit bigger. Four by four and you have lots of ones and zeros, that can be easy to get things mixed up. So just try to keep everything in the right order. Take your time and write out your steps. Okay? So when we're doing multiplication of matrices, this isn't a term by term kind of operation like it is when we're adding and subtracting. Okay, what we're gonna do is we're gonna take one row of the first matrix A. We're gonna multiply it element wise by a column in the second matrix matrix B. Okay, so term by term of that row in the column and that's gonna go in the corresponding row and column of our new matrix. Ok. Alright. So starting with the first row, first call. Okay, so our first row here from A and r. First column from Okay, we're gonna take the first element of the row which in this case is one, we're gonna multiply it to the first element in the column, which is one. Okay. And then we're gonna add, what do we add? We're going to move across the row and down the column and do the next term. So the next term is zero in our row times zero in our column. Okay then we add move on to the next term, zero in the row, zero in the column. And moving on to the final term and again in this case it's zero times zero. Alright, so this is our first row times the first column which we put in the first row, first column of our resulting matrix OK. We can do the same. Moving down so let's move down to road to And stick with column one. Well we're going to get here again the first element in the row, zero times the first element in the column one. Next element in the row, zero times the next element in the column zero. Okay. Plus third element in the row. Third element in the column zero times zero plus last element in the row, one last element in the column zero. Alright so now that we see how this works, let's go ahead with the rest of our terms. Okay so moving to the third row, we're gonna have zero times one. Last one Time 0 zero times zero plus zero times zero. And in the final row first column zero times 1 Plus zero Times 1. Oops, sorry zero times 0 plus one time zero Plus zero times 0. Okay now for the other columns, for the sake of space, I'm not going to write every multiplication. I'm just gonna write the result of the multiplication. Okay so we're into the second column, first row, we're going to have one time zero. It's going to be zero plus zero times zero, 010 times zero, zero plus zero times one. Also zero. Okay, Same column so column to row 20 times zero is zero, zero times zero, 00 times zero is zero and one times one is one. So zero plus zero plus zero plus one. We're working And still column two. Now on to row three, we get zero times zero gives us 01 times zero, gives us 00 times zero, gives us zero and zero times one gives us zero. Final row, column 20 times 000 times 001 times zero is zero and zero times one is zero. Okay, so again 0.0. Okay, third column, first row we get one times zero is 00 times 100 times zero is 00 times zero is zero. Okay, so this is just going to be a zero, 3rd column, 2nd row zero times 000 times 100 times zero is 01 times 00. Okay, so all zeros there and again, I'm just not writing out every step just for the sake of space. But it's the exact same process that we did in this first column. That's why we showed it all out in this first column. So that you could see the pattern. Okay, third row, third column zero times 001 times 110 times 000 times 00. Okay, so we get one there. Final row, third column zero times 000 times one is 01 times zero is 00 times zero is zero. Final row, final final not final row, first row, final call one time 000 times 000 times 100 times 00. Okay, so zero there, second row, last call zero times 00 times 00 times 11 times zero. Those are all zero, third row, last column zero times 01 times 00 times 10 times zero. Those are all zero. And the final row, final column zero times 00 times 01 times 10 times zero. Ok, so we did get a one there. All right now we've done the hard part? We've done the multiplication now. We just need to do a little bit of arithmetic to get this final matrix. Okay, so let's give ourselves some more space here and we find that the product A. B. Okay. The first row, first column we have 10. So we get one. Okay, The next working down the column we get zero, zero and zero in the next column. 0100. And we already have our final two columns. 00100001. Okay, what you'll see is that the product A. B is actually equal to the identity matrix? In four dimensions I four. Alright, great. So we've done a B. Now let's move on to be A. We've written out all of those steps when we did A B. So we're not going to write them all out again for B. It's the same process. Okay So we're gonna write out just the um results of the terms to save ourselves some space. Make it a little bit cleaner but we're using the same process. Okay, so B times A. B. Is 1000 in the first column, 0001 in the second column, 0100 in third column and 0010 in the fourth column. Okay. Times A 1000 in the first column, 0010 in the second column. 0001 in the third column. 0100 in the fourth column. Alright, and what is this going to equal? Well let's do it. We are going to get the first row. Okay. First call one times one is one. Hey, zero times 000 times and zero times 00. Okay, sticking with the same column. Working down to row two. Okay, zero times 100 times 001 times 000 times 00. Okay and again we're working across the row at the same time, we work down the column. Okay, now we're on row three. First element of the row, zero times the first element of column one. That gives us zero, ok, zero times zero zero times 01 times zero. All zeros there. Okay. Final row, first column, zero times 101 times 000 times 000 times 00. Okay, we're done. The first column. We move across to the second column. We start back with row 11 time zero zero zero time 000 times 100 times 00. Okay, moving on to the second row. Second column zero times 000 times 001 times 110 times 00, third row. Second column, zero times 000 times 000 times 10 and one times 00. Okay, Final row, second column zero times 001 times times 10 and zero times 00. Alright, third column back to the first row, one time 000 times 000 times 00 and zero times 10. Okay. Second row zero times 000 times times 000 times 10, third row, third column zero times 000 times 000 times 00 and one times 11. Last row, third column zero times 001 times 000 times 00 and zero times 10. Final column back to the top row, one times zero is 00 times one is 00 times zero is zero and zero times zero is zero. Okay, Second row zero times 000 times one zero one time 000 times 00, third row. Final column zero times 000 times 100 times 001 times 00. Final row. Final column zero times 001 times 110 times 000 times 00. Okay, so again, we're starting with matrices with a lot of zero. So we're gonna end up with a lot of zeros in our resulting matrix. Okay, We're done the hard part. We're done. The multiplication. We just need to add up all of these terms. Okay and what we're going to get is 1000 in the first column. 0100 in the second column 0010 in the third column 0001 in the last column. Which again will notice is the identity matrix and four dimensions I four. Okay. Alright. So we worked out a B and B. A. Let's go back up and answer the question. Okay. Now the question is asking us to use what we found about A B and B. A. To show if B is the multiplication inverse of A. Okay. What we found was that a B was equal to B. A. And they are both equal to the identity matrix in four dimensions. Okay. And this means because we have this that B is the multiplication inverse of A. Okay, so we have a B. Is the multiplication inverse of a. Thanks everyone for watching. I hope this video helped see you in the next one.

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