# Linear Algebra Problems

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 1st matrix x the 2nd matrix give the same answer as the 2nd matrix x the 1st matrix?

5.  Find the matrix product C x D.

6.  DT is given by .  What is CT?  (C is called a symmetric matrix.)

7. (-2) * DT is given by .  What are 3 * C and (-4) * CT?  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 = -13i - 1j +5k.  What are the CROSS Products?

v x u?

v x w?

u x w?

is u x v = v x u?