Linear Algebra and Its Applications, exercise 1.4.21

Exercise 1.4.21. An alternative way to compute the matrix product AB is as the sum

$c_1r_1 + c_2r_2 + \cdots + c_nr_n$

where $c_i$ is the ith column of A, $r_i$ is the ith row of B, and the product $c_ir_i$ is a matrix.

1. Provide an example showing the procedure above for a 2×2 matrix.
2. Show that the above procedure gives the correct answer for $(AB)_{ij} = \sum_{k=1}^{k=n} a_{ik}b_{kj}$

Answer: (a) We choose the following 2×2 example matrices, with product as shown:

$\begin{bmatrix} 1&2 \\ 3&4 \end{bmatrix} \begin{bmatrix} 3&5 \\ 7&9 \end{bmatrix} = \begin{bmatrix} 17&23 \\ 37&51 \end{bmatrix}$

Using the alternative mechanism above we can also compute the product as

$\begin{bmatrix} 1&2 \\ 3&4 \end{bmatrix} \begin{bmatrix} 3&5 \\ 7&9 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \begin{bmatrix} 3&5 \end{bmatrix} + \begin{bmatrix} 2 \\ 4 \end{bmatrix} \begin{bmatrix} 7&9 \end{bmatrix} = \begin{bmatrix} 3&5 \\ 9&15 \end{bmatrix} + \begin{bmatrix} 14&18 \\ 28&36 \end{bmatrix} = \begin{bmatrix} 17&23 \\ 37&51 \end{bmatrix}$

(b) For an mxn matrix A and nxp matrix B, the kth column of A and the kth row of B are as follows:

$c_k = \begin{bmatrix} a_{1k} \\ a_{2k} \\ \vdots \\ a_{nk} \end{bmatrix} \quad r_k = \begin{bmatrix} b_{k1}&b_{k2}&\cdots&b_{kn} \end{bmatrix}$

and their matrix product is

$c_kr_k = \begin{bmatrix} a_{1k}b_{k1}&a_{1k}b_{k2}&\cdots&a_{1k}b_{kn} \\ a_{2k}b_{k1}&a_{2k}b_{k2}&\cdots&a_{2k}b_{kn} \\ \vdots&\vdots&\ddots&\vdots \\ a_{nk}b_{k1}&a_{nk}b_{k2}&\cdots&a_{nk}b_{kn} \end{bmatrix}$

so that we have $(c_kr_k)_{ij} = a_{ik}b_{kj}$. If we define $C = c_1r_1 + c_2r_2 + \cdots + c_nr_n = \sum_{k=1}^{k=n} c_kr_k$ then we have

$C_{ij} = \sum_{k=1}^{k=n} (c_kr_k)_{ij} = \sum_{k=1}^{k=n} a_{ik}b_{kj}$

so that C = AB as hypothesized.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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