Matrices of the same dimentions can be added together element-wise.
Multiplying a matrix by a scalar multiplies every element of the matrix by that scalar.
For two matrices A,B, for AB to be defined, the number of columns in A must be the same as the number of rows in B.
If A is m×n and B is n×p, then AB will be m×p.
If you lay the ith row of A over the jth column of B, multiply element-wise, and add the results together, you get element i,j of AB.
Ex.
ABAB=(213−5)=(413−263)=(2(4)+3(1)1(4)+(−5)(1)2(3)+3(−2)1(3)+(−5)(−2)2(6)+3(3)1(6)+(−5)(3))=(11−101321−9)
For a matrix A, its transpose AT is the matrix whose columns are the rows of A.
Ex.
AAT=(acbd)=(abcd)
If A,B,C are matrices, r,s are scalars, and Im is an identity matrix:
A+B(A+B)+CA+0r(A+B)(r+s)A(rs)AA(BC)A(B+C)(B+C)Ar(AB)ImA(AT)T(A+B)T(AB)T=B+A=A+(B+C)=A=rA+rB=rA+sA=r(sA)=(AB)C=AB+AC=BA+CA=(rA)B=A(rB)=A=AIm=A=AT+BT=BTAT.
Note that AB=BA is not necessarily true! (Matrix multiplication is not commutative.)
An identity matrix Im is a m×m matrix with 1s along the diagonal and 0s everywhere else.
Ex.
I4=⎝⎜⎜⎛1000010000100001⎠⎟⎟⎞
This acts as an analogue to 1 for matrices – it is the multiplicative identity.