Adding and Subtracting Matrices

Definition:

Given two matrices A and B of order (m x n)

one can add or subtract these matrices by adding or subtracting each of the respective entries of the matrix.

As we can see from this example the order of the two matrices is (m x n). The resulting matrix also has an order of (m x n). Two matrices of equal order can be added/subtracted because each entry of one has a corresponding entry in the other. In general, a matrix of order (m x n) can only be added/subtracted to another matrix of order (m x n), i.e., the number of rows and the number of columns in both matrices must be equal.

Properties of Addition:

• Commutative Law - For two matrices A and B of equal order:

A+ B = B + A

Again, based on algebraic properties, each component of the matrix obeys this law, thus it is a property of the matrix.

• Associative Law - For two matrices A and B of equal order:

A + B + C = (A + B) + C = A + (B + C)

Just as in scalar addition and subtraction, the answer is not dependent on the order of addition.

• If O is the zero (null) matrix, then for any matrix A, of the same order:

O + A= A+ O= A

One can see that adding zero to the respective entries of a matrix does not change the matrix A.

Properties of Subtraction:

• Rules for addition may apply for subtraction if subtracting a matrix may be equivalately treated as adding the opposite sign, i.e., A - B = A + (-B). The principle of subtraction should be viewed instead as the adding of the negative of the matrix, i.e., A - B = A + (-B).

Examples:

1) Let A and B represent matrices of order (2 x 5) as follows

Add the matrices together:

2) Let R and S represent two matrices whose entries are functions of a variable t. If:

and

Subtract matrix S from matrix R: