## Linear Algebra and Its Applications, exercise 1.4.4

Exercise 1.4.4. Compute the number of multiplications required to multiply an mxn matrix A by an n-dimensional vector x. Also compute the number of multiplications required to multiply A by an nxp matrix B.

Answer: The result of multiplying A (which is mxn) by x (which is nx1) is an mx1 column vector b. Each entry of b is the inner product of a row of A (containing n entries) with all the n entries of x, and therefore requires n multiplications. Since b has m entries, the total number of multiplications is thus m times n or $mn$.

More formally, for each entry $b_i$ of b we have $b_i = \sum_{j = 1}^{j = n} a_{ij}x_j, \: { \rm for } \: i = 1, \ldots, m$

Each sum requires n multiplications, and there are m sums in toto.

When multiplying A times B, the result is the mxp matrix AB. Each entry of AB is the inner product of a row of A (containing n entries) and a column of B (also containing n entries), and requires n multiplications. Since AB contains m times p entries, the total number of multiplications is therefore m times p times n or $mnp$.

More formally, for each entry $c_{ij}$ of B we have $c_{ij} = \sum_{k = 1}^{k = n} a_{ik}b_{kj}, \: { \rm for } \: i = 1, \ldots, m \: { \rm and } \: j = 1, \ldots, p$

Each sum requires n multiplications, and there are m times p sums in toto.

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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### 1 Response to Linear Algebra and Its Applications, exercise 1.4.4

1. Mus-haf Akil says:

thank uuuuuuu very much