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MATRIX ALGEBRA
MGT 4850
Spring 2009
University of Lethbridge
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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.
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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
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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 ⎦
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Also example of vector multiplication!!!
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Vector Multiplication
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Vector Multiplication???
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Matrix Multiplication NotCommutative or Cancellative
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Identity matrix
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Linear Systems as a Matrix Product
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Ax=b
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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
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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)
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Matrix Inverses
• Let A be a square matrix. Then a (two-sided) 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.
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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.
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Examples of Inverses
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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.
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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
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Basic Properties of Determinants
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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),
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Example
• Use the Cramer’s rule to solve the system
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Solution
• The coefficient matrix and right-hand-side vectors are