Matrices 8.0-8.2 Quiz 1

Properties of Transpose

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

Types of Multiplication?

1. Scalar 2. Matrix Multiplication

Special Matrices

1. Zero Matrix (additive identity) 2. Additive Inverse Matrix 3. Square Matrix - Triangular Matrices (Upper Triangular/Lower triangular) 4. Diagonal Matrix 5. Scalar Matrix 6. Identity Matrix (multiplication identity) 7. Symmetric Matrix

Important properties

1. associative (AB)C = A(BC) 2. distributive A(B+C) = AB + AC (B+C)D = BD + CD

Adding and Subtracting Matrices

1. can only happen when 2 matrices are same dimension 2. add corresponding entries 3. commutative (A+B = B+A)

Matrix Multiplication

1. dont have to be same dimensions, but for ex in mXn and nXp, the column of the first and the row of the second must be the same 2. resulting matrix is mXp (AmXn * BnXp = CmXp) 3. Performed by multiplying the elements of each row of the first matrix by the elements of each column of the second matrix. Add the products. 4. not commutative

3 elementary functions

1. interchange any 2 rows 2. multiply any row by a nonzero constant 3. add a nonzero multiple of 1 to another row

Gaus-Jordan Elimination

1. perform gausian elimination 2. perform 3 elementary operations as many times as needed to turn every entry above each pivot to zero 3. all pivots are 1s

division of matrices is multiplication of inverse

A^-1(AX) = A^-1(b) I X = A^-1(b) analogous to 3x = 2 (1/3)3x = 2(1/3) x = 2/3

Zero Matrix

All entries are 0 ( A + 0 = 0 + A, leaves matrix unchanged)

Scalar Matrix

a diagonal matrix where all the entries along the main diagonal are the same, where Aij = 0 whenever i≠j and Aii = C

Square Matrix

a matrix with the same number of rows and columns ( AnXn)

What is a matrix?

a rectangular array of numbers where AmXn (m = number of row, n = number of columns)

Triangular Matrix - Lower Triangular

a square matrix is a lower triangular matrix if every entry above main diagonal is 0 where Aij = 0 whenever i<j

Identity Matrix (multiplication identity)

a square matrix is an identity matrix if it is a scalar matrix and every entry along main diagonal is 1, where In = [Aij]nXn and Aij = 0 whenever i≠j and Aii = 1 ex) 1. works like a scalar ( IA = A or BI = B)

Triangular Matrix - Upper Triangular

a square matrix is an upper triangular matrix if every entry below main diagonal is 0 where Aij = 0 whenever i>j

Diagonal Matrix

a square matrix where every entry off the main diagonal is 0, where Aij = 0 whenever i≠j

Form of Linear Equations?

a1x1 + a2x2 + a3x3 + .... + anxn = b a1 = coefficients x1 = variables b = constant

Scalar Multiplication

let O< be a scalar and A be an mXn matrix ( O<A = [ O<Aij ] mXn ) --> multiply each entry by alpha

how to perform gausian elimination

obtain augmented matrix and perform 3 elementary functions until row echelon form is reached

Division (transpose)

the rows of A are turned into the columns of A^T ( [1 2 3] --> [1 4] [4 5 6] [2 5] [3 6] )

Symmetric Matrix

when A^T = A

Additive Inverse Matrix

if A is an mXn matrix, then its additive inverse is -A --> A + (-A) = (-A) + A => 0mXn