Matrices Math 9

rows and columns:

A matrix with m row and n columns is an m x n matrix (read m by n) Note: A square matrix has the same number of rows as columns

Properties of Matrix Addition

A+B is a mxn matrix (Closure property of addition) A+B = B + A ( Commutative property of addition) There is an unique mxn matrix 0 such that 0 +A = A + 0 = A ( Associative property of addition) And identity and inverse (see above)

Properties of Matrix Multiplication

AB is a nxn matrix (Closure property) (AB) C = A(BC) (Associative property of multiplication) A (B+C) = AB + AC / (B+C) A = BA + CA (Distributive property) 0A = A0 = 0 (multiplicative property of zero)

The product of two matrices:

Can only be done if A has the same number of rows as B has of columns. Each element of AB is the sum of the products of the corresponding elements in the row of A and the column of B. Note: the multiplication of matrices is not commutative

Determinants:

Every square matrix with real number elements has a number associated with it - called a determinant. [ a b/c d] = ad - bc.

Additive identity and inverse of a matrix:

Identity: the 0 matrix (four zeros) Inverse:the matrix that you add to get 0

Multiplication Inverse Matrices:

Only square matrices have inverses. If A and B are square matrices and AB = BA = I then B is the multiplicative inverse of matrix A, written as A-¹ ex: inverse of [a b/c d] is 1/det A [ d -b/ -c a] solved.

Multiplicative Identity Matrices:

The multiplicative identity is a square matrix with it's number of rows the same as the number of columns given (ex: I₃ = (100/010/001)

Scalar products:

The product of a scalar, k, and a mxn matrix in which each element is equal to k time the corresponding element

equal matrices:

Two matrices are equal if and only if they have the same dimensions and are identical, element by element. Can solve by setting equal sign per row and solving as a system.

addition and subtraction of matrices:

add/ subtract per corresponding term

A matrix:

any rectangular array of terms called elements

Third order determinant:

a₁ [ b₂ c₂/ b₃ c₃] - b₁ [a₂ c₂/ a₃ c₃] + c₁ [a₂ b₂/ a₃ b₃]

How to solve Matrices using Row Operations:

options include: - switch any two rows - multiply a row by a constant - add two rows (and replace one of the two you added with that sum) - combine any of these two operations

The elements:

the elements in a matrix are arranged in rows and columns and are usually enclosed by brackets