MATRIX ALGEBRA
MGT 4850
Spring 2008
University of Lethbridge
Laws of Arithmetic
• Let A,B,C be matrices of the same size m × n, 0 the m
× n zero
• matrix, and c and d scalars.
• (1) (Closure Law) A + B is an m × n matrix.
• (2) (Associative Law) (A + B) + C = A + (B + C)
• (3) (Commutative Law) A + B = B + A
• (4) (Identity Law) A + 0 = A
• (5) (Inverse Law) A + (−A) = 0
• (6) (Closure Law) cA is an m × n matrix.
Laws of Arithmetic (II)
•
•
•
•
(7) (Associative Law) c(dA) = (cd)A
(8) (Distributive Law) (c + d)A = cA + dA
(9) (Distributive Law) c(A + B) = cA + cB
(10) (Monoidal Law) 1A = A
Matrix Multiplication
• Definition of Multiplication
2x − 3y + 4z = 5
as a “product” of the coefficient matrix
[2,−3, 4]
and the column matrix of unknowns
⎡x⎤
y│
⎣z⎦
Also example of vector
multiplication!!!
Vector Multiplication
Vector Multiplication???
Matrix Multiplication Not
Commutative or Cancellative
Identity matrix
Linear Systems as a Matrix Product
Ax=b
Laws of Matrix Multiplication
• Let A,B,C be matrices of the appropriate
sizes so that the following multiplications
make sense, I a suitably sized identity
matrix, and c and d scalars.
(1) (Closure Law) The product AB is a matrix.
(2) (Associative Law) (AB)C = A(BC)
(3) (Identity Law) AI = A and IB = B
Laws of Matrix Multiplication
(4) (Associative Law for Scalars) c(AB) =
(cA)B = A(cB)
(5) (Distributive Law) (A + B)C = AC + BC
(6) (Distributive Law) A(B + C) = AB + AC
• (skip from p.67 to p.101)
Matrix Inverses
• Let A be a square matrix. Then a (twosided) inverse for Invertible A is a square
matrix B of the same size as A such that
AB = I = BA. If such Matrix a B exists, then
the matrix A is said to be invertible.
• Application-if we could make sense of
“1/A,” then we could write the solution to
the linear system Ax = b as simply x =
(1/A)b.
Singular = nonivertable
Any nonsquare matrix is noninvertible. Square
matrices are classified as either “singular,”
i.e., noninvertible, or nonsingular,” i.e.,
invertible. Since we will mostly be
concerned with two-sided inverses, the
unqualified term “inverse” will be understood
to mean a “two-sided inverse.” Notice that
this definition is actually symmetric in A and B.
In other words, if B is an inverse for A, then
A is an inverse for B.
Examples of Inverses
Laws of Inverses
(1) (Uniqueness) If A is invertible, then it has
only one inverse, by A−1.
(2) (Double Inverse) If A is invertible, then
(A−1)−1 = A.
(3) (2/3 Rule) If any two of the three
matrices A, B, and AB are invertible, then
so is the third, and moreover, (AB)−1 =
B−1A−1.
Laws of Inverses
(4) If A is invertible, then (cA)−1 = (1/c)A−1.
(5) (Inverse/Transpose) If A is invertible,
then (AT )−1 = (A−1)T .
(6) (Cancellation) Suppose A is invertible. If
AB = AC or BA = CA, then B = C.
skip from p.103 to p.113
Basic Properties of
Determinants
Cramer’s Rule
• Let A be an invertible n×n matrix and b an
n×1 column vector.
• Denote by Bi the matrix obtained from A
by replacing the ith column of A
• by b. Then the linear system Ax = b has
unique solution x = (x1, x2, . . . , xn),
Example
• Use the Cramer’s rule to solve the
system
Solution
• The coefficient matrix and right-handside vectors are