2.4 Matrix Products

This is a rendition of a Möbius Strip by M.C. Escher

Matrix Multiplication

You were introduced to matrix multiplication last chapter recall the following:

If A is an mxn matrix thenB must be an nxp matrix or we can not find

the product ABIf B is an nxp matrix thenAB will be an mxp matrix

Matrix multiplication

There are 5 ways to multiply matrices that you will need to know for this course. We will look at 4 of them with this example.

1 0

4 5

2 –3

–1 3 .

Show for methods1) One entry at a time (row times column)2) A column at a time3) A row at a time4) A matrix times a column at a time (a column times a row)5) By partitions (this is not a good example

for that method)

Method 1:Multiply matrices one entry at a time

Take the dot product of each row in the first matrix with each column in the second matrix (learned last chapter)

1 0

4 5

2 –3

–1 3 .

Method 2 Multiply matrices a column at a time Look at the columns of the second matrix. the first column is 2, -1 that means take two

of the first column of the first matrix plus -1 times the second column of the first matrix this yields the first column of the answer matrix. Repeat this process for each column

1 0

4 5

2 –3

–1 3 .

Method 3Multiplying matrices a row at a timeLook at the rows of the first matrixThe first row of the first matrix is 1 0 this

means take 1 times the first row of the second matrix plus 0 times the second row. This is the first row of the answer matrix.

Repeat this process for each row.1 0

4 5

2 –3

–1 3 .

Method 4Columns times rows

Take column one of matrix A times row one of matrix B (the result is a matrix – this is much different than multiplying a row times a column)

Next take column two of matrix A times row 2 of matrix B (the result is a matrix)

Add the 2 matrices together to give the resultIf the matrix is larger continue this method

until each column has been multiplied by a row.

1 0

4 5

2 –3

–1 3 .

Example 3

Multiply the matrices using the given partition

Example 3

The Identity Matrix

I=

The identity matrix is a matrix that when multiplied by another matrix does not change the matrix AI=A or IA=A

Matrix Multiplication is not commutative!

(Except for a few special cases suchas multiplying by I or A-1)

AB≠BAbut

AI = IA = A (when A is square)Note: I is always a square matrix

Matrix Multiplication is associative

A(BC) = (AB)C

Many proofs and important results can (and will in the near future) be done in this class by simply moving parenthesis.

p. 77 1- 13 all (for 1-12 multiply by 4 methods)

A mathematician, standing puzzled at the photocopier and complaining to the secretary: "I set it to 'Single Sided Copy,' and now it comes out as a Möbius Strip!"