Consider the matrices A, B, C, and D below:

_{} _{} _{} _{}

1. Which matrices can be added to each other? If they may be added in more than one order, give both orders.

2. Which matrices can be subtracted from each other? If they may be subtracted in more than one order, give both orders.

3. Which matrices may be multiplied by each other? If they may be multiplied in more than one order, give both orders.

4. If matrices may
be multiplied by each other in more than one order, does the 1^{st}
matrix x the 2^{nd} matrix give the same answer as the 2^{nd}
matrix x the 1^{st} matrix?

5. Find the matrix product C x D.

6. D^{T} is
given by _{}. What is C^{T}? (C is called a symmetric matrix.)

7. -2 x D^{T} is given by _{}. What are 3 x C and
-4 x C^{T}? *Confirm your
result with MATLAB.*

**Example:** When
you find the inverse of a matrix using the method outlined in the slides, the
product of the diagonal elements of the original (left-side) matrix is the
determinant of the matrix. To find
det(C) and C^{-1}, we use the following procedure:

_{}

Step1: Subtract 2 x row 1 from row 2, and 3 x row 1 from row 3. We get

_{}

Step 2: Subtract 5 x row 2 from row 3. We get

_{}

Step 3: At this point, we know the determinant: 1 x (-1) x 18 = -18. To find the inverse, we must turn the left side matrix into the identity. We start by multiplying row 2 by -1 and dividing row 3 by 18. We get

_{}

Step 4: Now we subtract 5 x row 3 from row 2, and 3 x row 3 from row 1, getting

_{}

Step 5: To complete the process, we subtract2 x row 2 from row 1, getting

_{}. Thus C^{-1}
= 1/18 _{}, which you can check by multiplying C by C^{-1}, and
seeing whether you get the identity matrix.
Do this multiplication in MATLAB.

8. Find the determinant and inverse matrix using the method above for the matrices below.

_{} _{} _{} _{} matrix D above.

9. Consider the vectors u, v, and w below.

_{} _{} _{}

_{}; what are _{} Is _{}

10. Using the above vectors, we can compute u x v = -13*i* - 1*j*
+*5k*.
What is v x u? v x w? u x w?
Does u x v = v x u?

11. X = {2, 2, 3, 4, 5, 5, 6, 7, 8} and Y = (-3, -1, 1, 3, 5, 7, 9, 11}. The mean of X is 4.67, the median of X is 5, the range of X is 6, the mean deviation of X is 1.7, the sample standard deviation of X is 2.12, and the variance of X is 4.5. What are the corresponding quantities for Y?

12. The probability of each of the values in X is given by 2 or 5, 2/9 each; 3, 4, 6, 7, or 8, 1/9 each. The probability distribution function (starting at 2) has the values 2/9, 1/9, 1/9, 2/9, 1/9, 1/9, 1/9. The cumulative distribution function has the values (starting at 2) of 2/9, 3/9, 4/9, 6/9, 7/9, 8/9, 9/9. What are the corresponding values for Y?