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

MATRIX ALGEBRA

MGT 4850

Spring 2008

University of Lethbridge


Laws of arithmetic

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

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

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

Also example of vector multiplication!!!


Vector multiplication

Vector Multiplication


Vector multiplication1

Vector Multiplication???


Matrix multiplication not commutative or cancellative

Matrix Multiplication NotCommutative or Cancellative


Identity matrix

Identity matrix


Linear systems as a matrix product

Linear Systems as a Matrix Product


Matrix algebra

Ax=b


Laws of matrix multiplication

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 multiplication1

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

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

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

Examples of Inverses


Laws 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 inverses1

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

Basic Properties of Determinants


Cramer s rule

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

Example

  • Use the Cramer’s rule to solve the system


Solution

Solution

  • The coefficient matrix and right-hand-side vectors are


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