12 Chapter Test

Do you know HOW?

Find each sum or difference.

1. [ 4 7 − 2 1 ] − [ − 9 3 6 0 ] . matrix with 2 rows and 2 columns , row1 column 1 , 4 , column 2 7 , row2 column 1 , negative 2 , column 2 1 , end matrix . minus . matrix with 2 rows and 2 columns , row1 column 1 , negative 9 , column 2 3 , row2 column 1 , 6 , column 2 0 , end matrix
2. [ 4 − 5 1 10 7 4 21 − 9 − 6 ] + [ − 7 − 10 4 17 0 3 − 2 − 6 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 4 , column 2 negative 5 , column 3 1 , row2 column 1 , 10 , column 2 7 , column 3 4 , row3 column 1 , 21 , column 2 negative 9 , column 3 negative 6 , end matrix . plus . matrix with 3 rows and 3 columns , row1 column 1 , negative 7 , column 2 negative 10 , column 3 4 , row2 column 1 , 17 , column 2 0 , column 3 3 , row3 column 1 , negative 2 , column 2 negative 6 , column 3 1 , end matrix

Find each product.

3. [ 2 6 1 0 ] [ − 1 5 3 1 ] . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 6 , row2 column 1 , 1 , column 2 0 , end matrix . matrix with 2 rows and 2 columns , row1 column 1 , negative 1 , column 2 5 , row2 column 1 , 3 , column 2 1 , end matrix
4. 2 [ − 8 5 − 1 0 9 7 ] 2 . matrix with 2 rows and 3 columns , row1 column 1 , negative 8 , column 2 5 , column 3 negative 1 , row2 column 1 , 0 , column 2 9 , column 3 7 , end matrix
5. [ 0 3 − 4 9 ] [ − 4 6 1 3 9 − 8 10 7 ] . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 3 , row2 column 1 , negative 4 , column 2 9 , end matrix . matrix with 2 rows and 4 columns , row1 column 1 , negative 4 , column 2 6 , column 3 1 , column 4 3 , row2 column 1 , 9 , column 2 negative 8 , column 3 10 , column 4 7 , end matrix

Find the determinant of each matrix.

6. [ 1 0 0 0 1 0 0 0 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 0 , column 3 0 , row2 column 1 , 0 , column 2 1 , column 3 0 , row3 column 1 , 0 , column 2 0 , column 3 1 , end matrix
7. [ 2 3 0 − 1 1 0 4 2 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 2 , column 2 3 , column 3 0 , row2 column 1 , negative 1 , column 2 1 , column 3 0 , row3 column 1 , 4 , column 2 2 , column 3 1 , end matrix
8. [ 8 − 3 2 9 ] . matrix with 2 rows and 2 columns , row1 column 1 , 8 , column 2 negative 3 , row2 column 1 , 2 , column 2 9 , end matrix
9. [ 1 2 − 3 1 0 ] . matrix with 2 rows and 2 columns , row1 column 1 , 1 half , column 2 negative 3 , row2 column 1 , 1 , column 2 0 , end matrix

Find the inverse of each matrix, if it exists.

10. [ 3 8 − 7 10 ] . matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 8 , row2 column 1 , negative 7 , column 2 10 , end matrix
11. [ 0 − 5 9 6 ] . matrix with 2 rows and 2 columns , row1 column 1 , 0 , column 2 negative 5 , row2 column 1 , 9 , column 2 6 , end matrix
12. [ 3 1 0 1 − 1 2 1 1 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 3 , column 2 1 , column 3 0 , row2 column 1 , 1 , column 2 negative 1 , column 3 2 , row3 column 1 , 1 , column 2 1 , column 3 1 , end matrix
13. [ 1 1 2 2 1 3 2 1 1 ] . matrix with 3 rows and 3 columns , row1 column 1 , 1 , column 2 1 , column 3 2 , row2 column 1 , 2 , column 2 1 , column 3 3 , row3 column 1 , 2 , column 2 1 , column 3 1 , end matrix

Solve each matrix equation.

14. [ 3 − 8 10 5 ] − X = [ 2 8 − 1 12 ] . matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 negative 8 , row2 column 1 , 10 , column 2 5 , end matrix . minus x equals . matrix with 2 rows and 2 columns , row1 column 1 , 2 , column 2 8 , row2 column 1 , negative 1 , column 2 12 , end matrix
15. [ 3 2 − 1 5 ] X = [ − 10 − 11 26 − 36 ] . matrix with 2 rows and 2 columns , row1 column 1 , 3 , column 2 2 , row2 column 1 , negative 1 , column 2 5 , end matrix x equals . matrix with 2 rows and 2 columns , row1 column 1 , negative 10 , column 2 negative 11 , row2 column 1 , 26 , column 2 negative 36 , end matrix
16. 2 X − [ − 2 0 1 4 ] = [ 5 10 − 15 9 ] 2 x minus . matrix with 2 rows and 2 columns , row1 column 1 , negative 2 , column 2 0 , row2 column 1 , 1 , column 2 4 , end matrix . equals . matrix with 2 rows and 2 columns , row1 column 1 , 5 , column 2 10 , row2 column 1 , negative 15 , column 2 9 , end matrix

Find the area of each triangle with the given vertices.

17. vertices at (2, 3), ( − 3 , − 1 ) , open negative 3 comma negative 1 close comma  (0, 4)
18. vertices at ( − 2 , − 3 ) , open negative 2 comma negative 3 close comma  (5, 0), ( − 1 , 4 ) open negative 1 comma 4 close

Parallelogram ABCD has coordinates A(2, − 1 negative 1 ), B(4, 3), C(1, 5), and D ( − 1 , 1 ) . open negative 1 comma 1 close .  Write a matrix for the vertices after each transformation.

19. a dilation by a factor of 2 3 2 thirds
20. a translation 2 units right and 4 units down
21. a rotation of 270 ° 270 degrees
22. a reflection across y = x y equals x

Let u = 〈 − 2 , 1 negative 2 comma 1 〉, v = 〈1, 5〉, and w = 〈 − 1 , − 3 negative 1 comma negative 3 〉. Find each of the following.

23. u − v u minus v
24. 3v
25. 3 w + 2 u − 2 w 3 w plus 2 u minus 2 w
26. 3 v − 2 u 3 v minus 2 u
27. u · v u middle dot v
28. v · w v middle dot w

Determine whether each pair of vectors is normal.

29. 〈 3 , − 4 〉 , 〈 − 8 , 6 〉 left pointing angle bracket 3 comma negative 4 right pointing angle bracket comma left pointing angle bracket negative 8 comma 6 right pointing angle bracket
30. 〈 5 , − 2 〉 , 〈 3 , 4 〉 left pointing angle bracket 5 comma negative 2 right pointing angle bracket comma left pointing angle bracket 3 comma 4 right pointing angle bracket

Do you UNDERSTAND?

31. Open-Ended Write a matrix that has no inverse.
32. Writing Explain how to determine whether two matrices can be multiplied and what the dimensions of the product matrix will be.
33. Shopping A local store is having a special promotion where all movies sell at the same price and all video games sell at another price. Suppose you buy 5 movies and 4 video games for $97.50 and your friend buys 3 movies and 6 video games for $103.50. Write a matrix equation to describe the purchases. Then solve the matrix equation to find the price of a movie and the price of a video game.
34. Writing Describe the advantages and disadvantages of writing a vector in matrix form instead of component form.

Table of Contents