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Lecture 3

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University of Toronto Scarborough

Mathematics

MATA23H3

Kathleen Smith

Winter

Description

Deﬁnition: Let A = a i‘ ∈ M n,kR) and B = b ‘j ∈ M k,m(R).
The matrix product C = AB is the n × m matrix C = c ij
where
k
X
cij ai‘ ‘ja bi1 1jb + i2 2j a b ik kj= a ib j
‘=1
th th
(a is the i row of A and b ij the j column of B.)
Deﬁnition: The n×n identity matrix, denoted I (or I ) is deﬁned
n
by I = δ .
ij
Deﬁnition:IfA = a ij ∈ M n,nR), the elementsaii i = 1,2,··· ,n
are called the diagonal elements or the diagonal of A.
The trace of A, denoted tr(A), is
n
X
tr(A) = a iia 11 + a22 + ··· + a .nn
i=1
A square matrix with all zero elements, except possibly on the diag-
onal is called a diagonal matrix.
Deﬁnition: If A = aij ∈ M n,kR), then the matrix aj i ∈
M (R) is called the transpose of A and is denoted A .
k,n
T
Deﬁnition: Let A ∈ M n,nR). If A = A , then A is said to be
T
symmetric. If A = −A , then A is said to be skew-symmetric.
Properties of matrix multiplication (when deﬁned)
A(B C) = (AB)C
I A = A B I = B
A(B + C) = AB + AC (A + B)C = AC + B C
λ(AB) = (λA)B = A(λB) Properties of transpose (when deﬁned)
T T
(A ) = A
(A + B) = A + B T
(AB) = B A T T
Properties of trace (when deﬁned)
tr(A + B) = tr(A) + tr(B)
tr(λA) = λ tr(A)
tr(AT) = tr(A)
tr(AB) = tr(B A) Given the system of linear equations
a x +a x +··· +a x = b
11 1 12 2 1k k 1
a21 1+a 22 2·· +a x 2k k= b 2
.
.
.
an1 1a xn2 2· +a x =nk k n
Think of the coeﬃcienti of x as being a column vector and the sys-
tem as linear combinations of these column vectors. We can write as
the matrix equation Ax = b,
a11 a12··· a1k x1 b1
a21 a22··· a2k x2 b2
where A = , x = and b = .
. . .
. . .
a a ··· a x b
n1 n2 nk k n
A is called the coeﬃcient matrix of the system.
k
Any v ∈ R such that Av = b is called a solution of the system. To solve the system, we combine A and b into a matrix,
a11 a12 ··· a1k b1

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