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