![]() For mental computation, the row-column rule is probably easier to use than the definition. Next, use B – 2 A = B + (–2 A ): 7 5 1 4 0 2 3 5 3 2 1 4 3 8 10 4 7 6 7 B A − − − − = + = − − − − − − The product AC is not defined because the number of columns of A does not match the number of rows of C. ![]() Exercises 29–33 provide good training for mathematics majors. Outer products also appear in Exercises 31–34 of Section 4.6 and in the spectral decomposition of a symmetric matrix, in Section 7.1. Exercises 27 and 28 are optional, but they are mentioned in Example 4 of Section 2.4. Or, these exercises could be assigned after starting Section 2.2. A class discussion of the solutions of Exercises 23–25 can provide a transition to Section 2.2. Exercises 23–25 are mentioned in a footnote in Section 2.2. Exercises 23 and 24 are used in the proof of the Invertible Matrix Theorem, in Section 2.3. ![]() (The dual fact about the rows of A and the rows of AB is seldom needed, mainly because vectors here are usually written as columns.) I assign Exercise 13 and most of Exercises 17–22 to reinforce the definition of AB. ![]() 83 2.1 SOLUTIONS Notes : The definition here of a matrix product AB gives the proper view of AB for nearly all matrix calculations. ![]()
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