Addition of matrices
Addition of Matrices
Matrix addition is a fundamental operation in linear algebra, where two matrices of the same dimensions are added together by adding their corresponding elements. This operation is only possible when the matrices have the same number of rows and columns.
Properties of Matrix Addition
Matrix addition has several important properties:
- Commutative Property: The order of addition does not change the result. [ A + B = B + A ]
- Associative Property: The way matrices are grouped in addition does not change the result. [ (A + B) + C = A + (B + C) ]
- Identity Element: The zero matrix acts as an additive identity. [ A + 0 = A ]
- Inverse Element: For every matrix ( A ), there exists a matrix ( -A ) such that: [ A + (-A) = 0 ]
How to Add Matrices
To add two matrices, follow these steps:
- Ensure both matrices have the same dimensions.
- Add corresponding elements from each matrix.
Formula
If ( A = [a_{ij}] ) and ( B = [b_{ij}] ) are two matrices of the same size, their sum ( C = [c_{ij}] ) is given by:
[ c_{ij} = a_{ij} + b_{ij} ]
where ( c_{ij} ) is the element in the ( i )-th row and ( j )-th column of matrix ( C ).
Example
Let's consider two matrices ( A ) and ( B ):
[ A = \begin{bmatrix} 1 & 3 \ 5 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix} ]
To find ( A + B ), we add corresponding elements:
[ A + B = \begin{bmatrix} 1+2 & 3+4 \ 5+6 & 7+8 \end{bmatrix} = \begin{bmatrix} 3 & 7 \ 11 & 15 \end{bmatrix} ]
Table of Differences and Important Points
| Property | Description | Example |
|---|---|---|
| Dimensions | Matrices must have the same number of rows and columns to be added. | ( A_{2x2} + B_{2x2} ) is valid, but ( A_{2x2} + C_{3x2} ) is not. |
| Commutative | The order of addition does not matter. | ( A + B = B + A ) |
| Associative | Grouping does not affect the result. | ( (A + B) + C = A + (B + C) ) |
| Identity Element | Adding the zero matrix does not change the original matrix. | ( A + 0 = A ) |
| Inverse Element | Adding a matrix and its negative results in the zero matrix. | ( A + (-A) = 0 ) |
Practice Problems
Add the following matrices: [ \begin{bmatrix} 4 & 5 \ 7 & 2 \end{bmatrix} + \begin{bmatrix} 1 & 0 \ 3 & 4 \end{bmatrix} ]
Given matrices ( P ) and ( Q ): [ P = \begin{bmatrix} -1 & 2 \ 3 & 5 \end{bmatrix}, \quad Q = \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} ] Find ( P + Q ).
Verify the commutative property of matrix addition using matrices ( A ) and ( B ) from the example above.
By understanding and applying these principles, students can confidently perform matrix addition and solve related problems in their exams.