MA 265 Chapter Revies... Chapter 1 - Chapter 5

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Chapter 1

Matricies

A = [a b c; d e f; g h i] this is a 3x3 Matrix. 3 rows and three colums. The rows are seperated by a semicolen.

B = [a b c d; e f g h] this is a 2x3 Matrix. 2 row and 4 colums. Entry Asub(2,3) = g

- matricies are only equal if each corrisponding entry is the same.

- Matrix Addition = add the values of the corresponding matrix entries.

- Matrix Scalar Multiplication = r[a1 a2; a3 a4] = [ra1 ra2; ra3 ra4]

- Linear Combinations = c1A1 + c2A2 + ... + ckAk where c is a scalar and A is a matrix.

- Transpose Matrix(A^T) = say A = [a1 a2 a3 a4; b1 b2 b3 b4; c1 c2 c3 c4] the A^T = [a1 b1 c1; a2 b2 c2; a3 b3 c3; a4 b4 c4]

     notice that A   is a (3x4) matrix = 3 rows and 4 columns 
                 A^T is a (4X3) matrix ... the rows and the colums interchange.

Back to 2010 Fall MA 265 Momin

- Matrix Multiplication : C = a1ib1j + a2ib2j + .... + aipbip where the a components are of one matrix and the b components are of another. the i components are also the comonents of the row values of a and the j components are the column components of the matrix b. Remeber: when multiplying matricies one always multiplies row(a) x column(b). Also: if matrix A = (3*5) and B = (5*1) and C= A*B the C = (3*1) or... becuase the matrix A and B are complatible to be multiplied, the resulting matrix C is the row component of A and the column component of B.

- Some Properties

    1.  A(BC) = (AB)C
    2. (A+B)C = AC + BC
    3. C(A+B) = CA + CB

    1. (A^T)^T = A
    2. (A+B)^T = A^T + B^T
    3.  (rA)^T = rA^T

-Diagonal Matrix :for matrix A with components a..... aij = 0 for i not equal j ex. [1 0 0; 0 2 0; 0 0 5]

-Scalar Matrix : Diagonal matrix where all diagonal are equal ex. [4 0 0; 0 4 0; 0 0 4]

-Identity Matrix(In) : Diagonal matrix with all ones ex. [1 0 0; 0 1 0; 0 0 1] for A = (3x3) ex. [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1] for A = (4x4)

  A*In = A

-Upper and Lower Triangle : are a Symmertric Matrix(S) where A^T = A and/or aij = bij

-Skew Summetrix Matrix(Ak): if A^T = -A and/or aij=-bij Also: entries on Main Diagonal = 0, A=S+K

-Nonsingular : the matrix is invertible and has an inverse. (AB = BA = In)

-Singular : the matrix is not invertible and has no inverse.

 For Singular
  Ax = b         x = A^-1b   where A^-1 is the inverse of A
  (AB)^-1 = A^-1 B^-1
  (A^-1)^-1 = A
  (A^T)(A^-1)T = In