Page 60 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let A be a matrix. Show that if the function defined for matrices x by satisfies the linearity

24. property, then for any real numbers α and β and any matrices w and z.

Prove: If A and B are matrices, then .
25.

Describe three different methods for computing a matrix product, and illustrate the methods by
26. computing some product three different ways.

How many matrices A can you find such that
27.

for all choices of x, y, and z?

How many matrices A can you find such that
28.

for all choices of x, y, and z? .

A matrix B is said to be a square root of a matrix A if
29.

(a) Find two square roots of .

(b) How many different square roots can you find of ?

(c) Do you think that every matrix has at least one square root? Explain your
reasoning.

Let 0 denote a matrix, each of whose entries is zero.
30.

(a) Is there a matrix A such that and ? Justify your answer.

(b) Is there a matrix A such that and ? Justify your answer.

Indicate whether the statement is always true or sometimes false. Justify your answer with a
31. logical argument or a counterexample.