## 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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