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. 1 2 3 7/2 - 3 1/2A = 1 3 4 B = - 1/2 0 1/21 4 3 - 1/2 1 - 1/2

Matrices A and B for finding products AB and BA in college algebra.

Accepted Answer

Hey everyone welcome back in this problem. We're given two matrices, A and B. And were asked to work out the products, A. B and B. A. Okay. In order to show if B is the multiplication of inverse of A. Now the matrices were given our A three by three and B. Three by three. Okay. When we're doing matrix multiplication, we need the number of columns of the first matrix to equal the number of rows of the second matrix. Ok. So in this case they have the same dimension. So that works out. Okay. And then if we want B to be the multiplication of inverse of A we would need the dimension of B to be exactly the same as the dimension of A. And again in this case they both have the same dimension. So that is a possibility. So we can go ahead and do the products and work out this problem. So let's start with the product A. B. Okay, so we're gonna have A. Which has first column negative 719, 2nd column negative 204. And third column 40 negative six A. Times B. First call of zero, three halves, one, second column 13 halves, five halves and third column half. Now, when we do the multiplication of two matrices, we don't do it element by element like we do with additional attraction. Okay, what we do is we take the first row of the matrix, gay or a row of our first matrix and multiply it element by element by a column of the second matrix. Okay. And we're gonna put that in the position of the corresponding row and column in our resulting matrix. So what does this actually look like when we do it? So let's start with the first row of a. Okay. Times the first column of B. And we're going to multiply element by element So we're gonna take the first element in the row negative seven, multiply it by the first element in the column zero. Okay then we're gonna add we're gonna move across the row and down the column. The next element in the row is negative to the next element in the column is 3/2 and then adding. And we're gonna do the same for the last element of the last element in the row, four times the last element in the column one. Yeah. And that is going to be our entry because we're using the first row. First column, that's our entry in row one, column one of our resulting matrix. Already, we can do the same thing. Moving down. Okay, so let's move onto row two. We're gonna stick with column one and what we get we're gonna do the same thing. So we get the first element of the row one times the first element of the column zero. Okay. Plus the next element in the row zero comes the next element in the column. The rehabs. Okay. Plus next element in the row zero. Next element in the column one. And again that was the second row of a. Times the first column of B. So we put that in row two, column one of our resulting matrix. We'll do this once more and green for this column K. So we're moving on to the third row, we're sticking with the same column column one of B. K. First element of the row nine times first element of the column zero plus first or sorry, next element in the row four times next element in the column three halves plus next element in the row negative six times next element in the column one. Okay. Alright so we worked out the first, call them of our matrix now we're gonna move on to the second column and again because we have three by three matrices when we multiply them. We're also gonna get a three by three matrix out of it. Alright, so in the next column we're gonna start back at row one and work through column two. So we're gonna have the first element in row one negative seven. Okay. Times the first element in the column one. Okay we're going to add the next element in the row negative two times the next element in the column three halves. Okay And then we're gonna add the last element in the rho times the last element in the column. Black house. Same thing moving down. Okay. Second row. Second call them one times one plus zero times three halves plus zero Times 3/2. Oops, Sorry 5/2. Bye. Alright, last row. Second column. First element in the row nine. First element in the column one multiply them and then add the next element in the row four. Next element in the column three halves AD. And then do the same for the last element negative six times five hats. Okay. Alright, so we've done the second column. Let's move on to the third column. They were just going to write out the results for the sake of space here. So in the first row we have negative seven times zero. Okay, so that's just going to be zero, negative two times one. It's going to be negative two. Okay, and then four times a half, that's going to be plus two. Moving down to the next row one time zero, that's going to be zero zero times one. Also going to be zero and zero times a half zero. Okay, so just zero is there? And then the last row, last column nine times zero, that's gonna give us 04 times one. It's going to give us one and negative six times one half is gonna give us -3. Okay, so now we've done the hard part. We've done the multiplication. Now, all we need to do is a bit of arithmetic within this matrix we found in order to get the answer. Alright, so let's give ourselves a bit more space here and we're just gonna work out this A B. Okay, so the first row. First column negative seven times zero? That's zero, negative two times three halves. That's gonna be negative three plus four. That's gonna give us one. Okay. Second row, first column Well each term is multiplied by zero, so that's just going to give us zero and the last row, first column the first term is going to be zero OK, four times three halves, it's gonna give us six and then we have minus six. So we're gonna get zero there as well. Moving on to the first row. Second column negative seven times one is negative seven negative two times three halves. Well that's gonna be negative three. So we have negative seven minus three. So we have negative 10 and then we have four times five halves. Well that's going to give us 10, so negative 10 plus 10 is going to be zero. Second row. Second column we have won the other two terms have zero in them, so we're just gonna get one there. Okay? Um still in the second column. Now we're on the final row. Nine times one gives us 94 times three halves. It's going to give us six. So nine plus six, gives us 15 and then we have minus six times five halves. Which is gonna give us minus 15. 15 minus 15. 0. Now top right? So third column, first row we have negative two plus two. That zero, third column second row we just have all zeros. And in that final spot, four minus three gives us one. What we see is that A. B. Is actually the identity matrix I. three. Okay so the identity matrix and three dimensions. Alright. So we worked out the product A. B. Let's move on to the product B. A. Give ourselves some more space to work here and we're going to work out the product B. A. Now in this case our matrix B. Okay. Same matrix B as before. First column 03 halves one, second column one, 3/2 5/5 and third column 011 half. Okay, times a. First column negative 719. 2nd column negative 204 and third column 40 negative six. Okay, so we're gonna use the exact same process as before. Okay. Four. When we did A times B we're gonna work across one row in the first matrix and down a column in the second matrix. So first column first row we have zero times negative seven plus one times one Plus zero times 9, sticking with the same column moving down to row two, we get three halves times negative seven. Okay. Plus 3/2ves times one plus one, times nine. And we're gonna stick with that same column. We're gonna go to row three. We're going to the first thing in the row, one times the first thing in the column -7. Okay, we're gonna add the next thing in the row 5/2 to the next thing in the column one And then add again the final thing in the row one half times the final thing in the column nine. Okay, So now we've dealt with column one. Let's move over to column two. Mhm. Again you can do this in any order. If you find it easier to go across the rows first and then Sorry? Yeah, across the road first and then down to different columns, you can do that. Alright, so first row, second, column zero times negative two plus one. Time zero Plus zero times 4. Okay, so corresponding element from the row we're choosing and the column we're choosing, working across the road and down the column. 2nd row, second column three halves times negative two, 3/2, time zero And one times 4. 3rd row. 2nd column one times negative two Plus 5/2 time zero Plus 1/2 times four. Alright, now we're onto the last column, starting with the first row, zero times four Plus one times 0 plus zero times negative six. 2nd row 3/2, Times four plus three halves, time, zero plus one, Time is -6. And last row last column one times 4. Let's fire paths time zero plus one half, that is negative six. Alright, so we've done the multiplication and hopefully now that you've had quite a bit of practice with those two matrices, you're starting to see that pattern in that process. Okay you just have to do it over and over again in order to practice. Okay so working this up we're gonna get in the top left, we have zero Plus one Plus 0. Okay working down the column, we have negative seven times three halves. This is gonna be negative 21/2 plus three halves plus nine. And in the 3rd row first column we get -7. Last five halves Plus 9/2. Moving across to the second column in the first row we just have zero plus zero plus zero. Next column we get negative three plus zero plus four. And then in the last row negative two Plus zero Plus 2. Okay. Final column 000. So we just have zero plus zero plus zero. Okay, second row, third column three halves times four gives us six plus zero minus six. And final row, final column we get four plus zero minus three. Okay, if we work this out we're gonna get one 00. The first column 010 and the second column and 001 in the third column. And you'll see that this is also equal to the identity matrix in three dimensions. Okay, Alright, so let's go back up. Okay, we found B. A. Is the identity matrix in three dimensions? We found A B. Is the identity matrix in three dimensions. And what we want to write is whether B is a multiplication inverse of A. Okay, so we found that a B is equal to B. A. And they are both equal to the identity matrix and three dimensions. And this means that B is the multiplication inverse. Okay. Okay. So the answer to this question is going to be B. Or a? Sorry, B is the multiple inverse of A. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 1 - 12, find the products AB and BA to determine whether B is the multiplicative inverse of A. 1 2 3 7/2 - 3 1/2A = 1 3 4 B = - 1/2 0 1/21 4 3 - 1/2 1 - 1/2

Key Concepts

Matrix Multiplication

Multiplicative Inverse of a Matrix

Identity Matrix

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