Matrix algebra
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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)

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MATRIX ALGEBRA

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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 ⎦

    [x


Also example of vector multiplication!!!


Vector Multiplication


Vector Multiplication???


Matrix Multiplication NotCommutative 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 (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.


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-hand-side vectors are


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