Chapter 6 - Matrices

matrices with n rows and m columns is a what matrix

n x m matrix

what is n

rows

what is m 

columns

square matrix 

number of rows = number of columns

zero matrix

all the elements are zero

identity matrix

square matrix in which the elements on the leading diagonals starting top left are all 1 and remaining elements are 0

what are identity matrices denoted by

where k describes size

what is the condition for adding and subtracting matrices

they must be same size

conditions for matrix multiplication

you can only multiply if the number of columns in first matrix = number of rows in second

does the order you multiply matrices matter 

yes as AB ≠ BA

how to multiply matrices

multiply elements in each row in first matric by corresponding eleements in each column in the 2nd matrix and add together

what is the size of multiplied matrix

number of rows in first and columns in second

determinant in 2 × 2 matric

ad - bc

type of matric if determinant is equal to 0

singular matrix

what type of matrix if determinant ≠ 0

non singular

which type of matrix has no inverse

non singular

how to find determinant of 3 × 3 matrix

times a b c by its minor

minor of element in 3 x e matric 

determinant of the 2 × 2 matrix that remains after the row and column containing the element have been crossed out

imverse of matrix 

what is the matrix multiplied by its inverse equal to

identity

how to find inverse of 2 × 2 matric

transpose of matric 

rows become columns and columns become rowas 

how to find inverse of 3 × 3 matric

• find determinant

• matrix of minors

• alternating signs change

• transpose

• multiply by 1/det

what is (AB)^-1 equal to

B^-1 A^-1

how to work out x y z

when is a linear equation consitent

if there is at least one se of values that satisfies all the equations simultaneously

matrix meet at one point

• consistent

• one solution

• matrix is non singular so inverse exists

matrix sheaf

• consistent

• infinetly many solutions

• singular matrix

  • equations all need to be consistent but original equations not multiples of each other

matrix prism

• inconsistent equations

• no solutions

• singular

• originals and not any type of multiples

matrix parallel planes

• inconsistent

• no solutions

• singular

• at least 2 of the lhs of equations are multiples of eachother

matric same plane

• conssitent

• infinitely many solutions

• singular

• if all 3 equatrions of planes are multiples of eachother

self inverse matrix

matrix multiples by itself is identity matrix