Remember when adding or subtracting matrices, the matrices must be the same size.  Corresponding elements are added or subtracted to give elements of the new matrix.

Example: 
         1  3  5
A =   5  7  8
         2  0  9

         -2  8  0
B =    3  1   1
          5  0   2

                1-2   3+8  5+0       -1  11  5
A + B  =  5+3  7+1  8+1  =    8    8  9
                2+5  0+0  9+2        7    0  11

For those having difficulty calculating the amount of tip to leave at a restaurant, here's a quick and easy way. If you want to leave 15%, take 10% and then half of that 10% and add them together.

Total bill : $24

10% = 2.40
+ 5%   1.20
Tip =   $3.60

If you wish to leave 20% and you don't feel like multiplying, just take 10% and double it.

10% is simple to get. Just move the decimal point over 1 to the left. 10% of 25.78 is 2.58 (rounded off).

When multiplying matrices, remember the number of rows in the first matrix must equal the number of columns in the second matrix.  See how this multiplication occurs in the example below.

If the number of rows in the first matrix does not equal the number of columns in the second matrix, then the matrices cannot be multiplied. See the example below.

Matrix A has 2 rows and matrix B has 3 columns. Therefore these matrices cannot be multiplied together.

Here's a tip for working with formulas involving summations. I was working with a student yesterday who was getting confused with squaring a sum of values and squaring each individual value and then summing. This caused numerous errors when calculating the value of the correlation coefficient.

If the formula asks for sum x^2 then you square each x value and then take the sum. 

For example, if the x values are 2, 3, 5, 6, 9 square each value first

2^2 = 4
3^2 = 9
5^2 = 25
6^2 = 36
9^2 = 81

Now get the sum of these values

4 + 9 + 25 + 36 + 81 = 155


If the formula asks for (sum x)^2, you get the sum of the x values, then square the result.  In the above data set, we get

2 + 3 + 5 + 6 + 9 = 25, then square 25 to get 625.

The key to remember is to perfom everything inside the parentheses first.  Recall the order of operations is

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

You can remember this by remembering the phrase "Please Excuse My Dear Aunt Sally"

Front and back cover artwork for book.

Continuing on the theme of probability, suppose you have 12 songs on a CD and you want to play them in random order. How many ways can those songs be played and what is the probability that the song order on the CD will be the order played?

The song order on the CD is only 1 possible way the songs can be played. The probability is then 1 divided by the number of possible ways all songs could be played.

Have 12 slots, each representing a song selection  ___  ____ ____ ___ ___ ___ ___ ___ ___ ___ ___ ___

In the first spot, any of the 12 songs could be played, then in the second spot any of the remaining 11 songs could be played, in the next spot any of the remaining 10 songs could be played and so on.  Therefore the possible ways the songs could be played is 12 * 11 * 10 * 9 ..... * 1.  This is represented as 12!.  The probability is then 1/12!

Suppose you flip a fair coin (probability of heads = probability of tails) 6 times and the coin lands "tails" face up each time.  What is the probability that the next toss will also be a tail? What is the probability that the next toss is a head?

One might think that it's more likely to get a head since the first 6 tosses were tails, but in fact the probability that the next toss is a tail is .5, as is the probability that the next toss is a head.  The coin has no memory, so to speak. Even if the first 100 tosses landed tails up, the next toss still has a 50 percent chance of landing tails up.

Now the probability that all 6 coins tossed landing tails up is (1/2)(1/2)(1/2)(1/2)(1/2)(1/2) = 1/64.

Each toss of the coin is an independent event. What occurred previously on the coin toss has no bearing on the outcome of the next coin toss.