# Matrix Addition and Matrix Subtraction

Just like ordinary algebra, matrix algebra has operations like addition and subtraction.

## How to Add and Subtract Matrices

Two matrices may be added or subtracted only if they have the same dimension; that is, they must have the same number of rows and columns.

Addition or subtraction is accomplished by adding or subtracting corresponding elements. For example, consider matrix A and matrix B.

A =
 1 2 3 7 8 9
B =
 5 6 7 3 4 5

Both matrices have the same number of rows and columns (2 rows and 3 columns), so they can be added and subtracted. Thus,

A + B =
 1 + 5 2 + 6 3 + 7 7 + 3 8 + 4 9 + 5

A + B =
 6 8 10 10 12 14

And,

A - B =
 1 - 5 2 - 6 3 - 7 7 - 3 8 - 4 9 - 5

A - B =
 -4 -4 -4 4 4 4

And finally, note that the order in which matrices are added is not important; thus, A + B = B + A.

Problem 1

Consider the matrices shown below - A, B, C, and D

A =
 1 2
B =
 3 5 4 6

C =
 4 5 6 6
D =
 -1 0 -2 0

Which of the following statements are true?

I.   A + B = C
II.  B + C = D
III. B - C = D

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

Solution

The correct answer is (C), as shown below.

B - C   =
 3 - 4 5 - 5 4 - 6 6 - 6

B - C   =
 -1 0 -2 0
=    D

Note that Matrices A and B cannot be added, because B has more columns than A. Matrices may be added or subtracted only if they have the same number of rows and the same number of columns.