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Sample Test : Matrices I

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides a sample test related to matrices.

Exercise 1

(9 points)

123456789−2abcdefghi=111111111123456789−2abcdefghi=111111111 size 12{ left [ matrix { 1 {} # 2 {} # 3 {} ## 4 {} # 5 {} # 6 {} ## 7 {} # 8 {} # 9{} } right ]` - 2` left [ matrix { a {} # b {} # c {} ## d {} # e {} # f {} ## g {} # h {} # i{} } right ]`=` left [ matrix { 1 {} # 1 {} # 1 {} ## 1 {} # 1 {} # 1 {} ## 1 {} # 1 {} # 1{} } right ]} {}. What are aa, bb, cc, dd, ee, ff, gg, hh, and ii?

Exercise 2

(9 points)

Matrix [A][A] is 420−6810420−6810 size 12{ left [ matrix { 4 {} # 2 {} # 0 {} ## - 6 {} # 8 {} # "10"{} } right ]} {}. What is A +A + 12AA+A+12A?

Exercise 3

(9 points)

Using the same Matrix [A][A], what is 2 12 A212A?

Exercise 4

(9 points)

46−53xy=02246−53xy=022 size 12{ left [ matrix { 4 {} # 6 {} ## - 5 {} # 3{} } right ]` left [ matrix { x {} ## y } right ]`=` left [ matrix { 0 {} ## "22" } right ]} {}. What are x x and y y?

Exercise 5

(9 points)

134n2567134n2567 size 12{ left [ matrix { 1 {} # 3 {} # 4 {} # n{} } right ]` left [ matrix { 2 {} ## 5 {} ## 6 {} ## 7 } right ]} {} =

Exercise 6

(9 points)

4−203n13682901−140424−203n13682901−14042 size 12{ left [ matrix { 4 {} # - 2 {} ## 0 {} # 3 {} ## n {} # 1{} } right ]` left [ matrix { 3 {} # 6 {} # 8 {} ## 2 {} # 9 {} # 0 {} ## 1 {} # - 1 {} # 4 {} ## 0 {} # 4 {} # 2{} } right ]} {}=

Exercise 7

(9 points)

3682901−140424−203n13682901−140424−203n1 size 12{ left [ matrix { 3 {} # 6 {} # 8 {} ## 2 {} # 9 {} # 0 {} ## 1 {} # - 1 {} # 4 {} ## 0 {} # 4 {} # 2{} } right ]` left [ matrix { 4 {} # - 2 {} ## 0 {} # 3 {} ## n {} # 1{} } right ]} {}=

Exercise 8

(9 points)

[ a b c d e f g h i ] [ some matrix ] = [ a b c d e f g h i ] [ a b c d e f g h i ][ some matrix ]=[ a b c d e f g h i ] . What is "some matrix"?

Exercise 9

(5 points)

[ a b c d e f g h i ] [ some matrix ] = [ c b a f e d i h g ] [ a b c d e f g h i ][ some matrix ]=[ c b a f e d i h g ] . What is "some matrix"?

Exercise 10

(8 points)

  • a. Write two matrices that can be added and can be multiplied.
  • b. Write two matrices that cannot be added or multiplied.
  • c. Write two matrices that can be added but cannot be multiplied.
  • d. Write two matrices that can be multiplied but cannot be added.

Exercise 11

(15 points)

  • a. Find the inverse of the matrix [ 4 x 1 -2 ] [ 4 x 1 -2 ] by using the definition of an inverse matrix.

    Note:

    If you are absolutely flat stuck on part (a), ask for the answer. You will receive no credit for part (a) but you may then be able to go on to parts (b) and (c).
  • b. Test it, by showing that it fulfills the definition of an inverse matrix.
  • c. Find the inverse of the matrix [ 4 3 1 -2 ] [ 4 3 1 -2 ] by plugging x = 3 x=3 into your answer to part (a).

Extra Credit:

(5 points) Use the generic formula for the inverse of a 2x2 matrix to find the inverse of [ 4 x 1 -2 ] [ 4 x 1 -2 ]. Does it agree with your answer to number 11a?

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