To add or subtract two matrices, their dimensions must be the same.
In scalar multiplication, we simply multiply every element by the scalar value: [abcd]×λ=[λaλbλcλd]
3. Matrix-Matrix Multiplication
1) Matrix-Vector Multiplication
We map the column of the vector onto each row of the matrix, multiplying each element and summing the result. [abcd]×[xy]=[ax+cybx+dy] An m x n matrix multiplied by an n x 1 vector results in an m x 1 vector.
2) Matrix-Matrix Multiplication
We multiply two matrices by breaking it into several vector multiplications and concatenating the result. [abcd]×[wxyz]=[aw+cxbw+dxay+czby+dz]
4. Matrix Multiplication Properties
Matrices are not commutative:
A×B ≠ B×A (Except B is an unit matrix)
Matrices are associative:
(A×B)×C=A×(B×C)
5. Inverse and Transpose
The inverse of a matrix A is denoted A−1. Multiplying by the inverse results in the identity matrix.
AA−1=A−1A=I
The transposition of a matrix is like rotating the matrix 90° in clockwise direction and then reversing it.