What is the Commutative Property of Matrix Addition? | Printable Summary

The Commutative Property of Addition says we can add numbers in any order and still get the same value
'a' and 'b' are variables that stand for real numbers
We can also add matrices in any order and still get the same answer
'a', 'b', 'c', and 'd' represent the elements in the first general matrix
'e', 'f', 'g', and 'h' represent the elements in the second general matrix

1. The order in which we add numbers does not matter
2. 'a' and 'b' are variables that stand for real numbers
1. An 'element' of a matrix is a value in the matrix
2. To add matrices, just add the elements in corresponding positions
3. Then make a new matrix with the same dimensions, and put each sum in its corresponding spot
1. An 'element' of a matrix is a value in the matrix
2. To add matrices, just add the elements in corresponding positions
3. Then make a new matrix with the same dimensions, and put each sum in its corresponding spot
4. Notice that even though we added in a different order, we still got the same matrix we did before!
1. So in our example it seems the order in which we add matrices doesn't matter
1. It didn't matter what order we added the matrices in the example
2. But can we add any two matrices in any order?
1. If we represent the numbers of the matrices as variables, we can get a generalized formula
2. An 'element' of a matrix is a value in the matrix
3. 'a', 'b', 'c', and 'd' represent the elements in the first matrix
4. 'e', 'f', 'g', and 'h' represent the elements in the second matrix
1. Two matrices are equal if they have the same dimensions and all their corresponding elements are equal
2. Look at the corresponding elements of our new matrices:
3. a+e=e+a
4. b+f=f+b
5. c+g=g+c
6. d+h=h+d
7. This looks like the Commutative Property of Addition we recalled earlier!
8. Since all the corresponding elements are equal, the two matrices must be equal