Matrix addition and subtraction are fundamental operations in linear algebra. These operations are performed element-wise between matrices of the same dimensions. Let's explore how these operations work with a simple example of adding two 2x2 matrices.
Let's examine the element-wise addition process step by step. We add corresponding elements from each position: 2 plus 5 equals 7, 3 plus 1 equals 4, 1 plus 2 equals 3, and 4 plus 3 equals 7. This gives us our final result matrix.
Matrix subtraction works similarly to addition, but we subtract corresponding elements instead. For example, 8 minus 3 equals 5, 6 minus 2 equals 4, 5 minus 1 equals 4, and 9 minus 4 equals 5. This gives us our final difference matrix.
Matrix addition and subtraction require matrices to have identical dimensions. Addition is both commutative and associative, meaning A plus B equals B plus A, and the order of grouping doesn't matter. However, subtraction is neither commutative nor associative.