2.4 matrix products this is a rendition of a möbius strip by m.c. escher
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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!"
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