Matrices Flashcards
Dimensions of a matrix
Written as x X y
No. of rows x no. of columns
Matrix vs column vector vs row vector
A matrix has 2+ rows and columns
A column vector has one column
A row vector has one row
If the value of a variable is a matrix
Use bold, capital letters
Adding/subtracting matrices
Must be the same dimensions
Add/subtract the elements in corresponding positions
Scalar multiplication
Multiply everything in the matrix by the scalar
Square matrix
Has the same amount of rows and columns
Zero matrix
A matrix where all the elements are zero
Identity matrix
A square matrix where the diagonal from the top left is filled with 1s, every other value is 0
How to find an element in a multiplied matrix?
- Take the corresponding row from the first matrix
- Take the corresponding column from the second matrix
- Multiply first in row by first in column and so on
- Sum the answers
What must be true to be able to multiply matrices
Columns in matrix 1 = rows in matrix 2
Does AB = BA
No
Dimensions of a multiplied matrix
Rows in a x columns in b
Determinant
Effectively the inverse of a function, multiplying by the determinant is the same as multiplying by the original
How to write determinant of matrix A
det(A) or |A|
If det(A) = 0
A is a singular matrix and doesn’t have an inverse
If det(A) is not equal to 0
A is non-singular and does have an inverse
Determinant of matrix a b
c d
ad-bc
Determinant of matrix a b c
d e f
g h i
a | e f | - b | d f | + c | d e |
| h i | | g i | | g h |
Do the 2x2 determinant of matrices not in the same row/column as the factor
The minor of an element in a 3x3 matrix
The determinant of the matrix remaining when the row/column it is in are eliminated
Inverse of a matrix
Written as M^-1, undoes the effect of a matrix
MM^-1 = M^-1M = I
Inverse of a 2x2 matrix
- ( d -b)
———-
det(A). ( -c a)
What must you do when multiplying by an inverse
Do it to the front of both sides of the equation or the back of both
Write that MM^-1 = I before removing
A^T
Transpose of matrix A, inverted rows and columns
Inverting a general matrix
1) find det(A)
2) make a matrix of minors
3) use that matrix with each second sign reversed
4) transpose it
5) multiply by 1/det(A)
Using matrices to solve simultaneous equations
Write the coefficients of x.y and z in a matrix with the top row the first equation ensuring the order (x,y,z)
That multiplied by column vector (x y z) gives you the column vector of the answers
Inverse the 3x3 matrix
Multiply that inverse by the answers to find x.y,z
Write out and apply to context if necessary
What can you do for matrix simultaneous equations?
Use your calculator as long as you write the output of each step
Matrix simultaneous equations when you have to form your own
State what each variable is
Find the equations from the information, leave 0 as a coefficient if one only contains 2 variables
write as ax + by + cz = d
Repeat the steps from when you are given equations
What do 3 simultaneous equations with 3 unknowns form?
3 planes in 3d
Consistent
If there are solutions that satisfy all of the equations
First step to find the geometric representation of an equation
If the coefficients and answer one equation are all those of another raised by a factor then those two are the same (consistent with infinitely many solutions)
If the coefficients are raised by the same factor but not the answer then those two are parallel (inconsistent - no solutions)
Else: continue
Second check to find the geometric representation
Non-singular: one unique solution, consistent
Singular: inconsistent or infinitely many solutions, continue
What to do for the geometric representation if singular
Eliminate one variable by substitution
If one is a multiple of the other it is a sheaf, consistent, infinitely many solutions
If not it is inconsistent with no solutions, a prism
