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Lecture 13

# Lecture 13.pdf

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University of Waterloo

Mathematics

MATH 136

Robert Sproule

Winter

Description

Monday, February 3 − Lecture 13 : Matrix multiplication.
Concepts:
1. matrix multiplication.
2. multiplyingtwo matrices.
3. block-multiplying two matrices.
13.1 Definition of “matrix multiplication” between two matrices Aand B of dimension
m × n and n × k, respectively.
Operation What we start with : What we get :
Matrix
multiplication
13.1.1 Examples − Compute the (2, 3) entry of the product of the given matrices D
and E.
Answer: 0
13.1.2 Example − Compute the (2, 3) th entry of the product E D for the matrices D
and E above.
Answer: 2
13.1.3 Definition − Let A n × n= [aij be a square matrix. The entries {a , 11, a 22… 33
a nn is referred to as being the main diagonalof A. The matrix A is called the
identity matrix of dimension n, if every element on the main diagonal of Ais 1 while
all others are zero. That is, a =ii for all i = 1 to n and a = ijwhenever i ≠ j. The
identity matrix is denoted by I or I n × n Note – Two matrices A m × nand Bs × kare said to be multipliable only if n snd s are
equal.
13.1.4 Example − Consider the matrix A and the two following identity matrices:
We see that I plays the role of the “multiplicative identity” in matrix algebra.
13.2 Rules of matrix algebra − We previously listed a few addition, scalar multiplication
and transposition properties in lecture 12. We now add a few more matrix algebra
properties involving matrix-multiplication. All of these can be proven directly from the
definition. The proofs are not difficult but rather tedious to write out. In this list we
assume that pairs of matrices which are multiplied are indeed multipliable.
a) (αβ)A = α(βA) e) C (A + B) = CA + CB
b) A(BC) = (AB)C f) (αA)B= A(αB)
T T T
c) I n × n n × mA, A n × m m × m A g) (A B) = B A
d) (A + B)C= AC + BC h) (αA) = αA T
13.3 Observation − Looking over the rules of matrix algebra above we see that operations
(addition, scalar multiplication and multiplication) on matrices are very similar to
operations on numbers and/or vectors. They “behave” in the way we expect them to and
appear natural. There are however two significant rules that do differ from what we might
expect:
1. Multiplication of matrices is not commutative; i.e., A B ≠ BA. (Since Aand B
m × n n × m
might be multipliable as AB but not in as BA since their dimensions don’t allow it).
Even if

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