matrices

matrix

a grid of elements contained within brackets, has many real-world uses

order of a matrix

for a matrix with m rows and n columns, the order is m x n

(for example m x n as in 2 x 2, not 4)

common matrix types

square matrix- number of rows and columns are the same

zero matrix- all elements are 0, denoted by 0

multiplying a matrix by a zero matrix always returns a zero matrix

identity matrix- a matrix where the elements on the \ diagonal are all 1 and all other elements are 0, denoted by I

multiplying a matrix by an identity matrix always returns the non-id matrix

adding and subtracting matrices

(only works for matrices of the same order)

add/subtract the numbers in corresponding positions

multiplying and dividing matrices by scalars

multiply/divide every element in the matrix by a scalar

multiplying matrices

for matrices A and B where A has n columns and x rows and B has n rows and y columns:

for i 1 to n:

multiply equivalent terms(same position from the left/top of the row/column) in Aci and Bri

add together the multiples

append to product to ABciri

the product matrix will have the number of rows of A and the number of columns of B

order of matrix multiplication

"for matrices A and B where A has n columns and B has n rows" does NOT hold vice versa- sometimes AB exists but BA doesn't

therefore AB is not equivalent to BA

determinant of a matrix

a function that takes a square matrix and outputs a number, denoted as det M or |M|

if det M = 0 M is a singular matrix

for a 2x2 matrix with columns a b and c d will have a determinant of ad - bc (\ - /)

determinant of a 3x3 matrix

-expand along any row or column

-find the minor for each item along it(the determinant of the matrix left over if you delete its corresponding row and column)

-multiply each item by its minor and append it to the equation(which sign to append it with depends on its position in the matrix- see the diagram)

determinant of a 3×3 matrix, visually

determinants of matrices with calculators

you can in fact do this thumbs up emoji

transpose of a matrix

the transpose of a matrix M, denoted as M^T is found by interchanging the rows and columns

inverting matrices

the inverse of a matrix M is M^-1 such that (M^-1)M = M(M^-1) = 1 (its negative reciprocal)

A^-1 = C^T/detA

inverse matrix rules

if M is singular there is no inverse matrix

if two matrices A and B are non-singular (AB)-1 = (A^-1)(B^-1)

to divide matrix A by matrix B, multiply A by B^-1

inverting 2×2 matrices

-switch the values in the \ diagonal

-change the signs of the values in the / diagonal

-divide by the determinant

inverting 3×3 matrices

for matrix A

-find the matrix of minors, M, by finding the minor for every element and creating a matrix of their values

-find the matrix of cofactors, C, by combining the matrix of minors with the matrix of signs

-transpose C to form C^T (this matrix is sometimes called the adjugate of A)

-find the determinant of A, det A

-divide C^T by det A to form A^-1

inverting matrices with calculators

you can in fact do this thumbs up emoji

solving systems of equations using matrices

three sets of linear equations in the form ax + by + cz = n can be expressed with:

a, b and c as rows in a 3x3 matrix, A

x, y and x in a 1x3 matrix, B

the values of n as columns in a 1x3 matrix, C

AB = C

B = (A^-1)C

solving systems of equations using matrices, visually

solutions of systems of equations

a system of linear equations is consistent if there is at least one solution, for consistent systems:

-if the matrix corresponding to the set of linear equations is non-singular, the system has one unique solution

-if the matrix is non-singular, there are infinitely many solutions