For each pair of matrices A and B, find (a) AB and (b) BA.

Question

$A = [\begin{matrix} - 1 & 0 & 1 \\ 0 & 1 & 1 \\ - 1 & - 1 & 0 \end{matrix}],$

Accepted Answer

Hello, everyone. We are going to find the value of A B and B A. For the following matrices, we have 23 by three matrices. One is labeled A and one is labeled B to begin with, we will multiply A B in that order. Recall that to multiply matrices, the columns of the first matrix must match the rows of the second matrix. And then since these are both three by three matrices, they do. So for A B, my first element will be the product of the first row of a multiplied by the first column of B. So I'm gonna multiply corresponding elements and add them together. Since there are a lot of zeros in my second matrix, I'm gonna kind of add them as I go versus writing them down. So first I will have negative four multiplied by one which is negative four. Next I would have zero multiplied by zero, which is zero. And then the third elements. So four multiplied by zero is also zero. So the sum there is negative four, that's the first element in our first row of our solution matrix. The second element is from the first row of A and the second column of B. So first row of a negative four multiplied by zero, which is the first element in the second row of B is zero. Next, we have zero multiplied by one which is zero. And we're going to add to that four multiplied by zero, which is zero. So our second element here is zero. For the third element, we do the first row of a multiplied by the third column of B. So negative four multiplied by zero is zero zero, multiplied by zero is 04, multiplied by one is four. So four plus all those zeros is four. So we have our first row done. Second row will come from the first element in it will be the second row of a multiplied by the first column of B. So the first element in the second row of A is zero, multiplied by one is zero. Next, we would have four multiplied by zero, which is zero. And we'd add four multiplied by zero, which is still zero. So we have another zero. So the next element in that row is the second row of a multiplied by the second column of B. So the first element zero multiplied by zero is zero, four, multiplied by one is four and then four multiplied by zero is zero. So we end up with four as that element. For the third element of that row, we will do the second row of a multiplied by the third column of B. So zero, multiplied by zero is zero four, multiplied by zero is zero four, multiplied by one is four. So we have a four onto the third row of our solution. So we will do the third row of a multiplied by the first column of B for the first element in our solution. So we have negative four multiplied by one which is negative four. And then we have negative four multiplied by zero, which is zero and then zero multiplied by zero, which is zero. So the element here is negative four. Next, we do the third row of a multiplied by the second column of B. So negative four multiplied by zero is zero, negative four, multiplied by one is negative four zero, multiplied by zero is zero. So this will be negative four. And for the final element in that solution matrix, we will do the third row of a multiplied by the third column of B. So we will have negative four multiplied by zero, which is zero, negative four, multiplied by zero again, still zero and zero multiplied by one which is also zero. So this element is zero. So our solution for A B is the top row is negative 404. The second row is 044, the third row is negative four and negative 40. Next, we're gonna do B A. Now I need to switch the order of these matrices. So that way I can follow it a little easier. I need to write B before A or I get confused. So I'm rewriting matrix B and then rewriting matrix A. This way I can see which one is my first matrix and which one is my second matrix. And then we will go through and do the same step. So for B A, the first element of the first row will be the first row of B multiplied by the first column of A. And again, there are a lot of zeros. So I'm gonna kind of keep a running total in my head. So the first row of B I have negative one multiplied by or sorry, one multiplied by negative four, which is negative four. And since the next two elements are zeros, I know that's gonna be a zero. So I'm gonna have negative four as my first element in the first row of my solution matrix. The next element in that row will be the first row of B multiplied by the second column of A. And again, I can kind of check my zeros. So my first element will be multiplied by A zero, give me zero. And then the rest of the first row of B is zeros. So that'll also be zeros. So our second element is zero to find the third element, it will be the first row of B multiplied by the third column of A. So one multiplied by four is four. And then again, the next two elements are going to be multiplied by zero. So I know they are zeros. So this will be four for the second row, we'll do the second row of B multiplied by the first column of A for the first element. And again, if I match up my zeros, my first element of the second row is a zero. So that'll be a zero answer. Second element is multiplied by zero. Third element will be multiplied by zero. So the answer in that element is zero for the next element, I have the second row of B multiplied by the second column of A. So I have zero multiplied by zero, still 01 multiplied by four. So that's four and then zero multiplied by negative four, which is zero. So we will have a four in this spot. For the third element of that row, we have the second row of B multiplied by the third column of A. So zero, multiplied by four is 01, multiplied by four is four and zero multiplied by zero is zero. So adding that up, we have four on to my third row in the third row of B multiplied by the first column of A will give us our first element in our third row of our solution. So zero multiplied by negative 400, multiplied by 00, one multiplied by negative four is negative four. So our answer there is negative four. Next, the third row of B multiplied by the second column of A zero, multiplied by zero is 00, multiplied by 401 multiplied by negative four is negative four. Finally, we'll do the third row of B multiplied by the third column of A. And the first two elements in the third row are zeros. So those will be zeros and one multiplied by zero is zero. So this will also be a zero. So here's our solution matrix for B or yeah, for B A and in the end, we actually ended up with identical matrices for A B and B A, they both come out to three by three matrix matrices where the first row is negative 404. The second row is 044 and the third row is negative four, negative 40. Have a nice day.

For each pair of matrices A and B, find (a) AB and (b) BA.
$A = [\begin{matrix} - 1 & 0 & 1 \\ 0 & 1 & 1 \\ - 1 & - 1 & 0 \end{matrix}],$

Key Concepts

Matrix Multiplication

Order of Multiplication in Matrices

Dimension Compatibility

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