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

Question

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

Accepted Answer

Hello, everyone. We are asked to find the value of A B and B A for the following matrices, we are given 23 by three matrices, one labeled A and one labeled B to begin with. I'm gonna multiply A B recall that when we multiply matrices, the order matters. So if we're working with A B A has to be our first matrix B has to be our second. And we know that since we are multiplying a three by three by a three by three, our answer will be a three by three matrix. The first element of the first row of the solution will come from the first row of a multiplied by the first column of B. There are a lot of zeros in these matrices. So we are going to keep a running total for each element in our head instead of writing it all down. So the first row of A, the first element is zero, multiplied by the first element of the first column of B which is one, so zero multiplied by one is zero. And then I noticed the next two elements of B are also zero. So this means that our first element of our solution matrix is zero. Our second element in that first row will be the first row of a multiplied by the second column of B. So we have zero multiplied by zero, which is 03 multiplied by one, which is three negative three multiplied by zero, which is zero. So the sum of that will be three. And our third element in that first row will be the first row of a multiplied by the third column of B. In the third column of B, I noticed the first two elements are zeros. So I'm gonna just worry about multiplying negative three is the third element of the first row of A by one. And that will give me negative three. Everything else will come out to zero. So negative three is the value for the second row. I'm gonna multiply the second row of a multiplied by the first column of B. So again, if I try to match up my elements ahead of time, I see that I have the first elements multiplied by A zero. The second elements multiplied by A zero as is the third. So that means my first element of my second row of my solution matrix is zero. Second element of the second row of the solution will be the second row of a multiplied by the second column of B. And again, the first elements are both zeros. Uh The second element three multiplied by one is three, third elements, both zero. So this will be a three for the third element of that second row, we do the second row of a multiplied by the third column of B. And so first element we have a zero and A zero. So that's 03, multiplied by zero is 01, multiplied by zero is zero. So this will be a zero for the third row of our solution matrix, we will do the third column, I'm sorry. The third row of a multiplied by the first column of B. So first elements I have a zero multiplied by a number. So that's zero. Second element zero, multiplied by zero is zero. Third element three multiplied by zero is zero. So our first element of our third row is also a zero. Second element will be the third row of a multiplied by the second column of B. So if the first elements, one of them is a zero. So that's zero. Second element zero times one is 03, multiplied by zero is zero. So this will also be a zero. Last element here will be the third row of a multiplied by the third column of B. So the first element is zero, multiplied by zero, which is zero, second is zero, multiplied by zero. Again, third is three multiplied by one. So that's three. So A B gives us the three by three matrix 03, negative three, second row 030, 3rd row 003. So now we need to find B A and before I find B A, I'm gonna rewrite my matrix, my matrices. So that B is listed first, I do this because otherwise I know I personally will lose track of which one was the first and which one was the second? My brain cannot read these in reverse. So it helps me to rewrite it and it might help you to do so also. So now that they're rewritten, I am gonna be a lot more comfortable multiplying B and A in that order again, since these are both three by three matrices, we know our, our answer is gonna be a three by three matrix. So the first element of the first row of our solution will be the first row of B multiplied by the first column of A. And notice that the first column is all zeros. So that means that this element we don't even have to do the multiplication anything multiplied by zero is zero. So if all of those are zeros, this will be a zero. So the second element of that first row will be the first row of B multiplied by the first or sorry. The second column of A. So we have three multiplied by one, which is three plus zero, multiplied by three is zero plus zero, multiplied by zero is zero. So we have a three here. Third element of that first row is the first row of B and the third column of A. So one multiplied by negative three is negative three. And then the next two are zeros. So we are going to say this is negative three. Our second row for A uh for B A will be the second row of B. And the first element comes from multiplying it by the first column of A first columb A is still zero. So this will still be zero. The next element will be the second row of B multiplied by the second column of A. So zero multiplied by three is 01, multiplied by three is 30, multiplied by 30 is zero. So this will be a three, the third element in that row will be the second row of B in the third column of A. So zero multiplied by negative three is 01 multiplied by zero is zero and zero multiplied by three is zero. So this element will be zero for the third row, we have the third row of B multiplied by the first column of A where everything is almost a zero there and uh A is all zero. So this will again be a zero. Our second element will be the third row of B multiplied by the second column of A and three multiplied by zero is zero, three, multiplied by zero is 01 multiplied by zero is zero. So this is zero. Our last element will be the third row of B multiplied by the third column of A. So zero, multiplied by negative three is zero zero, multiplied by zero is 01, multiplied by three is three. So hey, we got a value other than zero. Awesome. So this is our B A matrix. And if you notice they are both the same, so they are both three by three matrices where the first row reads 03 negative three. The second row is in the third row is 003. Those are our two solution matrice C have a nice day.

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

Key Concepts

Matrix Multiplication

Order of Multiplication in Matrices

Dimension Compatibility

Watch next

For each pair of matrices A and B, find (a) AB and (b) BA. A=[01−1010001],B=[100010001]A = \(\left\)[ \(\begin{matrix}\) 0 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)]

Key Concepts

Matrix Multiplication

Order of Multiplication in Matrices

Dimension Compatibility

Watch next

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