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

Question

For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7. A= , B =

Accepted Answer

Hello, everyone. We are asked to find the value of A B and B A for the following matrices. Matrix A is a two by two matrix with the first row of zero negative nine and second row of negative 36 B is also a two by two matrix. The first row is seven negative two. The second row is negative 18, we have four answer choices. Each has 22 by two matrices as the product for A B and B A starting with A B. Since these are both two by two matrices, I know that my result will be a two by two matrix. So starting with the first element of the first row, we will find this by multiplying the first row of matrix A by the first column of matrix B. So the first element of matrix A is zero. And we're gonna multiply that by the first element of the first column of matrix B which is seven, we're gonna add to that the second element of the first row of A which is negative nine multiplied by the second element of the first column of B which is negative one moving to the right for my second element in my first row, for my answer matrix, this will be the first row of A multiplied by the second column of B. So the first element of the first row is zero multiplied by negative two, which is the first element of the second column of B. And we're gonna add to that the second element of A from row one which is negative nine multiplied by the second element of the second column of B which is eight going down to the second row. I'm gonna multiply the second row of matrix A by the first column of matrix B. So I have the first elements negative three multiplied by the first element of the column which is seven plus the second element which is six multiplied by the second element of the column which is negative one final element. We need to plan out for this matrix. And this will be the second column of matrix A or sorry. The second row of matrix A multiplied by the second column of matrix B. So this will be negative three multiplied by negative two. We're gonna add to that six multiplied by eight. So now that I have all my values, I'm going to do the multiplication and addition. So I can find my final solution. So for the first element of the first row zero multiplied by seven is zero, negative nine multiplied by negative one is a positive nine So our first element in that row is a positive nine. Second element zero multiplied by negative two is zero and then negative nine multiplied by eight is negative 72. Down to the second row, I have negative three multiplied by seven, which is negative 21. And that's gonna be added to negative one multiplied by six. So that should be negative 27. And then the second element there, we have negative three multiplied by negative two, which is a positive six. And we're gonna add to that. The product of six and eight, which is 48. So positive six plus 48 is 54. So this is our solution matrix for A B A two by two matrix, first row is nine negative 72. 2nd row is negative 27 54. But we're not done as we also need to do B A. So for B I, I personally need to rewrite my B and A matrices. So that B is written first, if I don't do that, I have a hard time keeping track of which row and which column I'm working on. So I'm rewriting the matrixes or sorry, the matrices we were given. So now that I have them rewritten now I can work on my solution matrix. So the first element of the first row of our solution is the first row of B multiplied by the first column of A. So seven multiplied by zero plus negative two multiplied by negative three. The second element of that row will be the first row of B multiplied by the second column of A. So seven multiplied by negative nine plus negative two multiplied by six. Second row of my solution matrix is the second row of matrix B multiplied by the first column of matrix A. So negative one multiplied by zero plus eight multiplied by negative three. And then the second element of that second row is the second row of B multiplied by the second column of A. So negative one multiplied by negative nine plus eight, multiplied by six. All right, I have my values. So now let's multiply and add and get our solution. So zero times seven will be zero, negative two multiplied by negative three is six. So the first element in our solution is six. Uh next element going to the right, I'll have negative 63 plus negative 12. So that should be negative 75. First element in the second row, one multiplied by zero. I'm sorry, negative one multiplied by zero is zero. So eight multiplied by negative three is negative and then negative one multiplied by negative nine is a positive nine plus the product of eight and six which is 48 plus 48 is 57. So our solution matrix is again a two by two matrix. And this will be the top row is six, negative 75. The second row is negative 24 and matching that up, it is answer choice a have a nice day.

For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7. A= , B =

Key Concepts

Matrix Multiplication

Order of Multiplication

Dimension Compatibility

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