Chapter 10, Problem 12RCC

(a)

To determine

To define: the dimensions of A and B for the product AB.

Answer to Problem 12RCC

The dimensions of A and Bshould be equal to each other to obtain the product AB.

Explanation of Solution

The matrix product AB is defined only if their inner dimensions are equal to each other.

That is, the product is defined if the number of columns in A is equal to the number of rows in B.

(b)

To determine

To calculate: the product AB

Answer to Problem 12RCC

The product AB is calculated by multiply the two matrix A and B which will have the $n\times k$ dimension.

Explanation of Solution

If the product AB is defined, then the dimensions of A are $m\times n$ and the dimensions of B are $n\times k$ since the inner dimensions must match.

The product is a $m\times k$ matrix, where the each entry $u_{ij}$ is computed use the inner product of the $i^{th}$ row of A and the $j^{th}$ column of B.

Each row of A and each column of B is treated as a vector, allowing us to compute the inner product.

The inner product is defined as follows:

Given two vectors x and y both of length n, the inner product is the number obtained after compute $x_1{y}_1+x_2{y}_2\mathrm{...}+x_n{y}_n$ .

This is precisely why we the number of columns in A and the number of rows in Bis required to match-would not be able to compute an inner product for each entry in the new matrix otherwise