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Chapter 2: Problem 1

The sizes of matrices \(A\) and \(B\) are given. Find the size of \(A B\) and \(B A\) whenever they are defined. \(A\) is of size \(2 \times 3\), and \(B\) is of size \(3 \times 5\).

Short Answer

Expert verified
The product \(AB\) is defined and has a size of \(2 \times 5\). The product \(BA\) is not defined, so its size cannot be determined.

Step by step solution

01

Check the definition of \(AB\) and find its size

Since \(A\) is of size \(2 \times 3\), it has 2 rows and 3 columns. Similarly, \(B\) is of size \(3 \times 5\), so it has 3 rows and 5 columns. To check if \(AB\) is defined, we need to see if the number of columns of \(A\) (3) is equal to the number of rows of \(B\) (3). Since they are equal, the product \(AB\) is defined. Next, we find the size of the resulting matrix. To do this, take the number of rows of \(A\) (2) and the number of columns of \(B\) (5). So, the size of \(AB\) is \(2 \times 5\).
02

Check the definition of \(BA\) and find its size (if applicable)

Now, let's check if the product \(BA\) is defined. We need to compare the number of columns of \(B\), which is 5, with the number of rows of \(A\), which is 2. Since 5 does not equal 2, the product \(BA\) is not defined. To summarize: - The product \(AB\) is defined, and the size of \(AB\) is \(2 \times 5\). - The product \(BA\) is not defined, so its size cannot be determined.

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