Properties of Matrices - Addition, Multiplication, Transpose

Properties of Matrices - Addition, Multiplication, Transpose

TL;DR

The properties of matrices are the rules that govern the four core operations — addition, scalar multiplication, multiplication, and transpose — including which laws carry over from ordinary numbers and which break. This article covers each property with its formula, the big exception (matrix multiplication is not commutative: AB≠BA), and six worked examples that test the rules.

What Are The Properties Of Matrices?

The properties of matrices are the algebraic laws the four matrix operations obey: addition, scalar multiplication, multiplication, and transpose. Each operation has its own short list of rules, and together they tell you what you are allowed to do when rearranging a matrix expression.

Variable glossary. A,B,C denote matrices; O is the zero matrix (all entries 0); I is the identity matrix; k,l are scalars (ordinary numbers). The order of a matrix is its rows-by-columns count, and most properties require the orders to be compatible.

What Are The Properties Of Matrix Addition?

Matrix addition is the most well-behaved operation — it inherits every rule from ordinary number addition, as long as the matrices share the same order.

What Are The Properties Of Scalar Multiplication?

Multiplying a matrix by a scalar k scales every entry by k, and the operation distributes cleanly over both matrices and scalars.

The order m×n is preserved: scaling never changes a matrix's dimensions.

What Are The Properties Of Matrix Multiplication?

This is where matrices break from numbers, and the break is the most important single fact in the article.

Matrix multiplication is not commutative: in general AB≠BA.

The properties that do hold:

There is no general "multiplicative inverse" property: only some matrices have an inverse A^{-1} with AA^{-1}=I.

What Are The Properties Of The Transpose?

The transpose interacts with the other operations through four rules:

Examples Of Properties Of Matrices

Example 1

Verify A+B=B+A for A=[1 2; 3 4], B=[5 6; 7 8].

Add in each order, entry by entry:
A+B=[6 8; 10 12], B+A=[6 8; 10 12].
Final answer: Both give [6 8; 10 12], so addition is commutative.

Example 2

Show k(A+B)=kA+kB for k=2, A=[1 0; 2 1], B=[3 1; 0 4].
Left side: A+B=[4 1; 2 5], 2(A+B)=[8 2; 4 10].
Right side: 2A=[2 0; 4 2], 2B=[6 2; 0 8], 2A+2B=[8 2; 4 10].
Final answer: Both sides equal [8 2; 4 10], confirming the distributive property.

Example 3

Check whether AB=BA for A=[1 2; 0 1], B=[1 0; 3 1].
AB=[7 2; 3 1], BA=[1 2; 3 7]; therefore AB≠BA.
Final answer: AB=[7 2; 3 1] and BA=[1 2; 3 7]; they are not equal.

Example 4

Verify A(B+C)=AB+AC for A=[2 0; 1 3], B=[1 1; 0 1], C=[0 2; 1 0].
Left side: B+C=[1 3; 1 1], A(B+C)=[2 6; 4 6].
Right side: AB=[2 2; 1 4], AC=[0 4; 3 2], AB+AC=[2 6; 4 6].
Final answer: Both sides equal [2 6; 4 6], so the distributive law holds.

Example 5

Confirm (AB)^T=B^T A^T for A=[1 2; 3 0], B=[0 1; 4 2].
Ab=[8 5; 0 3], (AB)^T=[8 0; 5 3].
Now B^T A^T=[8 0; 5 3].
Final answer: Both equal [8 0; 5 3], so (AB)^T=B^T A^T.

Example 6

Show A+O=A and AI=A for A=[7 -2; 5 9].
A+O=[7 -2; 5 9] + [0 0; 0 0] = [7 -2; 5 9].
AI=[7 -2; 5 9] [1 0; 0 1] = [7 -2; 5 9].
Final answer: A+O=A and AI=A.

Why The Properties Of Matrices Matter

The properties of matrices exist to mark the boundary between the algebra of numbers, which students already know, and the algebra of matrices, which mostly looks the same but is not.

Key Takeaways