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Lecture 13

Lecture 13.pdf

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University of Waterloo
MATH 136
Robert Sproule

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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