Matrix operations are a fundamental concept in linear algebra, and understanding how to perform addition and subtraction of matrices is crucial for working with matrices in various fields such as mathematics, physics, engineering, and computer science. In this article, we will provide a comprehensive overview of matrix addition and subtraction, along with examples and a worksheet to help you practice these operations.
Matrix Addition and Subtraction: Basics
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. To add or subtract matrices, they must have the same dimensions, i.e., the same number of rows and columns.
Matrix Addition
Matrix addition is performed by adding corresponding elements in the two matrices. If we have two matrices A and B with the same dimensions, their sum is denoted as A + B. The elements of the resulting matrix are calculated as:
| Matrix A | Matrix B | Matrix A + B |
|---|---|---|
| a11 | b11 | a11 + b11 |
| a12 | b12 | a12 + b12 |
| a21 | b21 | a21 + b21 |
| a22 | b22 | a22 + b22 |
For example, let's add two 2x2 matrices:
| Matrix A | Matrix B |
|---|---|
| 1 2 | 3 4 |
| 5 6 | 7 8 |
Matrix A + B =
| Matrix A + B |
|---|
| 1+3 2+4 |
| 5+7 6+8 |
Matrix A + B =
| Matrix A + B |
|---|
| 4 6 |
| 12 14 |
Matrix Subtraction
Matrix subtraction is performed by subtracting corresponding elements in the two matrices. If we have two matrices A and B with the same dimensions, their difference is denoted as A - B. The elements of the resulting matrix are calculated as:
| Matrix A | Matrix B | Matrix A - B |
|---|---|---|
| a11 | b11 | a11 - b11 |
| a12 | b12 | a12 - b12 |
| a21 | b21 | a21 - b21 |
| a22 | b22 | a22 - b22 |
For example, let's subtract two 2x2 matrices:
| Matrix A | Matrix B |
|---|---|
| 1 2 | 3 4 |
| 5 6 | 7 8 |
Matrix A - B =
| Matrix A - B |
|---|
| 1-3 2-4 |
| 5-7 6-8 |
Matrix A - B =
| Matrix A - B |
|---|
| -2 -2 |
| -2 -2 |
Key Points
- Matrix addition and subtraction can only be performed on matrices with the same dimensions.
- The resulting matrix has the same dimensions as the original matrices.
- Matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)).
- Matrix subtraction is not commutative (A - B ≠ B - A).
Worksheet: Addition and Subtraction of Matrices
Practice the following problems:
- Add the following matrices:
| Matrix A | Matrix B |
|---|---|
| 2 3 | 1 4 |
| 5 6 | 7 8 |
- Subtract the following matrices:
| Matrix A | Matrix B |
|---|---|
| 10 12 | 3 4 |
| 15 18 | 7 8 |
- Add the following matrices:
| Matrix A | Matrix B |
|---|---|
| 1 2 3 | 4 5 6 |
| 7 8 9 | 10 11 12 |
| 13 14 15 | 16 17 18 |
What are the conditions for matrix addition and subtraction?
+Matrix addition and subtraction can only be performed on matrices with the same dimensions, i.e., the same number of rows and columns.
Is matrix addition commutative?
+Yes, matrix addition is commutative, meaning that the order of the matrices being added does not change the result (A + B = B + A).
Can matrix subtraction be performed on matrices with different dimensions?
+No, matrix subtraction can only be performed on matrices with the same dimensions.