Practice Problems 3

Lecture 3: Matrices and Matrix Operations

Problem numbers refer to the textbook (Chiang & Wainwright, 4th edition). These are not graded, but you should attempt them all before the quiz. Each part has its own Solution toggle: try it yourself first, then expand to check.

Exercise 4.2

1. Given A=[7−169],B=[043−2],C=[8361], find:

  1. A+B
Solution

A+B=[7397]

  1. C−A
Solution

C−A=[140−8]

  1. 3A
Solution

3A=[21−31827]

  1. 4B+2C
Solution

4B+2C=[01612−8]+[166122]=[162224−6]

2. Given A=[283051],B=[2038],C=[7263]:

  1. Is AB defined? Calculate AB. Can you calculate BA? Why?
Solution

AB is defined, since the number of columns in A is two, which equals the number of rows in B. AB=[2×2+8×32×0+8×83×2+0×33×0+0×85×2+1×35×0+1×8]=[286460138]

Not possible to calculate BA: B has two columns, but A has three rows.

  1. Is BC defined? Calculate BC. Is CB defined? If so, calculate CB. Is it true that BC=CB?
Solution

BC and CB are both defined, as both are 2×2. BC=[1446930]≠CB=[20162124]

4. Find the product matrices in the following (in each case, append beneath every matrix a dimension indicator):

  1. [020304230][800135]
Solution

[023620163]3×2

  1. [65−1104][4−15201]
Solution

[49343]2×2

  1. [35042−7][xyz]
Solution

[3x+5y4x+2y−7z]2×1

  1. [abc][700214]
Solution

[7a+c2b+4c]1×2