Matrix Operations
Matrices are rectangular arrays of numbers or other mathematical objects, for which operations such as addition, subtraction, and multiplication are defined. Below are the different types of matrix operations explained with examples.
1. Matrix Addition
Matrix addition involves adding corresponding elements of two matrices of the same dimensions.
Definition
If ( A ) and ( B ) are matrices of the same size, their sum ( C ) is given by:
where each element ( c_{ij} ) of ( C ) is:
Example
Let
The sum ( C ) is:
2. Matrix Subtraction
Matrix subtraction involves subtracting corresponding elements of one matrix from another of the same dimensions.
Definition
If ( A ) and ( B ) are matrices of the same size, their difference ( D ) is given by:
where each element ( d_{ij} ) of ( D ) is:
Example
Let
The difference ( D ) is:
3. Matrix - Matrix Multiplication
Matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second matrix.
Definition
If ( A ) is an ( m \times n ) matrix and ( B ) is an ( n \times p ) matrix, their product ( C ) is an ( m \times p ) matrix given by:
where each element ( c_{ij} ) of ( C ) is:
Example
Let
The product ( C ) is:
Example (3x3 Matrix)
Let
The product ( C ) is:
4. Scalar Matrix Multiplication
Scalar multiplication involves multiplying each element of a matrix by a scalar value.
Definition
If ( A ) is a matrix and ( k ) is a scalar, their product ( B ) is given by:
where each element ( b_{ij} ) of ( B ) is:
Example
Let
The product ( B ) is:
Example (3x3 Matrix)
Let
The product ( B ) is:
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