# Properties of Matrix Operations

Given that *A*_{mn}, *B*_{mn} and *C*_{mn} are matrices with m rows and n columns. If c and d are scalars, then the following properties are true.

(cd)*A* = c(d*A*)

**Commutative property of addition**

*A + B = B + A*

**Associative property of addition**

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

**Distributive property**

c(*A* + *B*) = c*A* + c*B*

(c + d)*A* = c*A* + d*A*

Here are some general rules about the three operations: addition, multiplication, and multiplication with numbers, called **scalar multiplication**.

**Properties involving Addition:**

Let *A*, *B* and *C* be *m×n* matrices. We have

1. *A*+*B* = *B*+*A*

2. (*A*+*B*) + *C* = *A* + (*B*+*C*)

3. *A+O = A, w*here *O* is the *m×n* zero-matrix (all its entries are equal to 0).

4. *A + B = O*, if and only if *B* = -*A*.

**Properties involving Multiplication:**

1. Let *A*, *B* and *C* be three matrices. If the products of *AB*, (*AB*)* C*, *BC*, and *A* (*BC*) are valid, then we have: (*AB*)*C* = *A* (*BC*)

2. If α and β are numbers, and A is a matrix, then we have:

*α (βA) = (α β)A*

3. If α is a number, and *A* and *B* are two matrices such that the product *AB* is valid, then we have

*α (AB) = (αA)B = A(αB)*

4. If *A* is an *n×m* matrix and *O* is a *m×k* zero-matrix, then we have:

*AO = O*

Note that *AO* is the *n×k* zero-matrix. So if n is different from m, the two zero-matrices are different.

**Properties involving Addition and Multiplication:**

Let *A*, *B* and *C* be three matrices. If the products of *AB*, *BC* and *AC* are valid, then we have:

(*A*+*B*)*C* = *AC* + *BC*

and

*A*(*B*+*C*) = *AB* + *AC*

If α and β are numbers, *A* and *B* are matrices, then we have:

*α (A+B) = αA + βB*

and

*(α +β) A = αA + βA*

**Example 1: **Calculate 4*C, AD*, *DA*, *BC*, 3*CB*, *C* (*A* + *B*), *AB*, *BA*, *CAD*, *DBC*, *AD* + (*CB*)^{T}, *DC* and *CD *for the given matrices below.

**Solution: **

4C = [20 12 24]

*DA* - The dimensions are not valid for multiplication (3 by 1 multiplied by 3 by 3). The inside dimensions do not agree.

*BC* - The dimensions are not valid for multiplication (3 by 3 multiplied by 1 by 3). The inside dimensions do not agree.

3CB = [342 213 180]

*C* (*A* + *B*) = [ 160 161 136 ]

*CAD* = 980

*DBC* - The dimensions are not valid for multiplication (3 by 1 *×* 3 by 3 *×* 1 by 3). The inside dimensions do not agree on either multiplication.

*CD* = 50

**Example 2: **What matrix would need to be added to *A* to produce the 3 by 5 zero matrix if

**Solution: **

The required matrix can be given as

**Example 3: **Given that:

Calculate:

*A + B*

*A – B*

*C + A*

**Solution: **

C + A: Not possible to calculate as both the matrices are of different dimension.