Matrix addition, subtraction and scalar multiplication

Matrices can be added or subtracted only when they have the same size. The relevant definitions:

To add or subtract two m x n matrices, add or subtract their corresponding entries.

The negative of a matrix is the matrix whose entries are the negatives of the original matrix.

A zero matrix is one of any size all of whose entries are 0. Notation: Omn if it's important to indicate the size, O if the size is unimportant or clear from the context.

 

The word "scalar" just means "number" – we use a separate word to distinguish numbers from other mathematical objects we want to calculate with, like matrices.

To multiply a matrix by a scalar, multiply every entry of the matrix by that scalar.

 

You can derive most of the calculation rules for matrix addition, subtraction and scalar multiplication from the rules for addition, subtraction and multiplication of ordinary numbers. Here is a list of ten rules that are considered to be "essential" for matrix addition and scalar multiplication.

Ten Essential Rules for m x n Matrices

Closure rules:
for m x n matrices A and B and any scalar c,

A + B is an m x n matrix

cA is an m x n matrix

Arithmetic rules for addition:
for any m x n matrices A, B and C,

A + B = B + A

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

There is an m x n matrix Omn such that A + Omn = Omn + A = A for all m x n matrices A.

For each m x n matrix A, there is an m x n matrix –A such that A + (–A) = O.

Arithmetic rules for scalar multiplication:
for any m x n matrices A and B and any scalars c and d,

c(A + B) = cA + cB

(c + d)A = cA + dA

(cd)A = c(dA)

1A = A

Why are these rules "essential"? After all, they're all pretty obvious – the last says that multiplying a matrix by the scalar 1 doesn't change the matrix, for example - and other obvious rules such as 0A = Omn are not on the list.

The point is, the rules in the list are very basic mathematical properties shared with other types of mathematical objects - it's the rules themselves that are important, not the objects that obey them. You'll see how all this works out later on in the course; for now, just make sure you understand what those rules say.

 

If you take an extended combination of sums and scalar multiples of matrices of the same size, you get what is called a linear combination.

A linear combination of several matrices A1, A2, ..., Ak of the same size is any matrix of the form

c1A1 + c2A2 + ... + ckAk

where c1, c2, ..., ck are all scalars.

For example, if A, B, C and D are all 2 x 3 matrices, then the 2 x 3 matrix 3A – 2B – 5C + D is a linear combination of A, B, C and D. So is 3(A + 2B) – 4C + 5(D – A + C) – B, since you can use the ten rules to simplify this expression to –2A + 5B + C + 5D.