Download Math1010 MAtrix

Matrix
an mxn matrix is a rectangular array of numbers, with m rows and n columns.
1 2
0 −1
a1,1
[ ⋮
am,1
2x2 matrix,
⋯ a1,n
⋱
⋮ ]
⋯ am,n
[
1 3
5 2
3
] 2x3 matrix
3
Addition
Add the 2 matrix together, element by element.
Scalar multiplication
You can multiply matrixes by scalar. Each element is multipled by the scalar.
Matrix multioplication
You can multiply an mxr by rxn to get mxn matrix.
the i,j entry eg AB comes from the ith row of A x jth column of B (summed)
Systems of equations
Ax=b
Where are is A is the a matrix of coefficients
x is a vector of variables, and b is a vector of the solutions for the variables.
Ax=0
0 is a the zero vector
it’s homogeneous
there is either infinite or unique solution.
matrix ops
Addition is commutative, A+B=B+A
Multiplication is not commutative, AB≠BA (some time it does, but it doesn’t always)
Addition is associative, (A+B)+C=A+(B+C)
Multiplication is associative, A(BC)=(AB)C –ASSCIAIVE LAW Says we can bracket multiplication
however we want.
Distributive law, multiplication distributes over addition, A(B+C)=AB + AC
Example: (A+B)2=A2+AB+BA+B2
Eros and Matrix multiplication
each elementary row op, corresponds to multiplying the whole system by a elementarily matrix.
Transpose
if A is a mxn matrix
AT is a nxm matrix
where (AT)ij=(A)ji
Properties:
(AT)T=A
(A+B)T=AT+BT
(AB)T=BTAT
Transpose reverses the order of a matrix product
Square – a matrix is square if m=n
A square matrix is symmetric if A=AT
A square matrix is skew symmetric if A=-AT. The downward rightward diagonal entries must be their
own negative .’. must = 0
Identity Matrix
A identity matrix (there are many depending on the size you use) you can multiply any matrix by and
1 0
you get the same matrix as before. Denoted by In where n the size. 𝐼2 = (
)
0 1
Inverse matrix
ONLY FOR SQUARE MATRIX
you can any matrix by it’s inverse and get an identity matrix.
AB=BA=In
Where A is a square matrix, B is it’s inverse.
eg
3 −13
9 1
where 𝐴 = (
),
𝐴−1 = (
) 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝐴 written
−2
9
2 3
3 −13 9 1
1
(
)(
)= (
−2
9
2 3
0
How to calculate
For 2x2 ONLY
1
𝑎 𝑏
𝑑
𝐴=(
),
𝐴−1 = 𝑎𝑑−𝑏𝑐 (
𝑐 𝑑
−𝑐
−𝑏
)
𝑎
If 𝑎𝑑 − 𝑏𝑐 = 0, 𝐴 ℎ𝑎𝑠 𝑛𝑜 𝑖𝑛𝑣𝑒𝑟𝑠𝑒
if 𝑎𝑑 − 𝑏𝑐 ≠ 0, A does have an inverse A-1
Things without inverses
0 0
0 0
On : 02 = (
) , 03 = (0 0
0 0
0 0
Any matrix with a row all zeros,
or a column all zeros
0
0) zero matrix
0
0
)
1
Any matrix with 2 equal Rows or scalar multiples of each other (rows).
Things with Inverses
In (has an inverse of In)
Elementarily matrix. (an elementary matrix is on that corresponds to an elementary row operation)
row swapping matrix is it’s own inverse
1 0 0
1 0 0
1
−1
scaling row operation marix A 𝐴 = (0 𝑎 0), 𝐴 = (0 𝑎 0),
0 0 1
0 0 1
Row adding matrix has an inverse that subtracts (taking you back to where you were before)
Properties
if A and B are nxn matrixes
A-1 is invertible A-1=A
AB is invertible (𝐴𝐵)−1 = 𝐵−1 𝐴−1
AT is invertible. (𝐴𝑇 )−1 = (𝐴−1 )𝑇
−1
Ak is invertible (𝐴𝑘 )
= (𝐴−1 )𝑘
A is a square matrix
A is invertible
E is any electuary matrix
if A is invertible the EA is also invertible.
if EA is invertible then A is invertible.
Doing elementary matrix operations doesn’t change wether a matrix is invertible.
A matrix is invertible iff it’s row echelon form is invertible.
Which means that every column of it’s echeloen form matrix is a basic column. (every leading entry
is along the diagonal)
A matrix to be invertable a matrix has to be made of emelentrairy matrixes