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

A=_{}, B=_{}, C=_{}, and D=_{}

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

**Example:** When
you find the inverse of a matrix using the method outlined in class (see the
review 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** for each of the following matrices using the method described
above.

A= _{} B= _{} C=_{}, and D=_{}

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

*u = _{} , v =_{} , and w = _{}*

* *

The DOT product of *u *and *v *is 1,
i.e.; _{}. What are
_{}

Is _{}

10. u x v = -13*i* - 1*j*
* +5k*.
What are the CROSS Products?

v x u?

v x w?

u x w?

is u x v = v x u?