Week 1 matrices Flashcards
3x2 matrix what is the no. of rows?
3 rows
2 columns
what is a column/ row vector?
c- only has 1 column
r- only has 1 row
what is the main diagonal?
the elements in the matrix, starting from the top left, then second row and second column
what is an identity matrix?
How do you find it using inverse?
where all elements along the main diagonal = 1
find it by: matrix A * inverse matrix A^-1 = identity matrix
what is a zero/null matrix?
matrix = 0
what is a skew-symmetric matrix?
where elements along the main diagonal = 0
P1
What does this mean for the determinant?
If matrix has identical rows then the determinant = 0
A+B?
SUBTRACT?
SUBTRACT- Same as addition but subtract corresponding no. instead
D-A?
Not possible as it’s 3x2 and 3x3
HAVE TO BE SAME SIZE
3D
Multiply all no. by multiplying factor outside matrix
2A - 3B
Multiply all the scalar factors with their respective matrix, then subtract with corresponding no.
Multiply +3 with matrix B, then leave the -ve last
What does transpose do?
Switches rows and columns
First column now = first row
What is a symmetric matrix?
Where if you try to transpose the matrix it = original matrix
E.G.
What does finding a trace of a matrix mean?
Trace A
Matrix multiplication
AC
1) Check if possible
ONLY WORKS WHERE: axb and cxd, where b and c must be same
2)Multiply A’s first row with C’s first column
What are the rules for multiplication for matrices?
1) Chain of multiplication must be same in photo (ABC order is sustained)
2) Multiplication is distributive
3) Transpose of a product = product of transposes multiplied in reverse order
4) Non-commutative- order of multiplication matters, AB doesn’t equal BA
Matrix A divided by Matrix B
A/B
What kind of matrices can we find the determinant of?
what is the notation for determinant?
Determinant pattern for low dimension matrices?
EX.
Determinant pattern for higher dimension matrices
Determinant properties?
How do you find the adjoint? (low dimension matrices)
1) swap 1,1 element with 2,2
2) swap 1,2 element with 2,1
3) use the sign pattern rule for matrices
4) then transpose the matrix
Find the adjoint (higher dimension matrices)
1) Start top left, and go along the row then start at left again on new row
2) for each number, cover its associated row and column and put in what’s left over (matrix) with determinant lines
3) repeat for all elements and apply sign pattern
4) find the determinant for each
5) transpose
(PHOTO)
What kind of matrix can the inverse be found for?
How do you find the inverse of a matrix? (formula)
Pattern for low dimension matrices?
- for square matrices (nxn)
What is the inverse matrix properties?
Find the inverse matrix for this:
how should this be expanded?
Terms in bracket should have Transpose, but now multiplied in reverse order
how should this be expanded?
Determinant of A transpose, is equal to the determinant of A
how should this be expanded?
how should this be expanded?ve
Then to get real solution, find actual determinants of both B and C, then put the numbers in a fraction.
how should this be expanded?e
Qd
1) (not included in the photo)
split the matrix, so C^-1 * D^-1
2) for each, Calculate the determinant (check if possible)
3) calculate the adjoint
4) inverse= (1/determinant) * adjoint
5) finally times the 2 inverses together
what kind of matrix is this?
symmetric matrix
what is a diagonal matrix?
non-zero elements in the main diagonal, with zero everywhere else
