Algebra of Matrices

Algebra of Matrices Important Points:

  • Matrices can be added or subtracted element-wise if they have the same dimensions.
  • Scalar multiplication is performed by multiplying each element of a matrix by a scalar value.
  • Matrix multiplication involves multiplying each row of the first matrix by each column of the second matrix.
  • The product of two matrices is only defined when the number of columns in the first matrix is equal to the number of rows in the second matrix.
  • Matrix multiplication is not commutative, meaning the order of multiplication affects the result. 

List of Important formulas to remember 

 

Algebra of Matrices:


  • Addition: $$A + B = [a_{ij}] + [b_{ij}] = [a_{ij} + b_{ij}]$$

  • Subtraction: $$A - B = [a_{ij}] - [b_{ij}] = [a_{ij} - b_{ij}]$$

  • Scalar Multiplication: $$kA = [ka_{ij}]$$

  • Matrix Multiplication: $$AB = [c_{ij}] \text{ where } c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$