Matrix multiplication doesn’t have commutative property (交换律). But some square matrices satisfy 𝐀𝐁=𝐁𝐀

Matrix multiplication has Associative property (结合律)

(𝐀𝐁)𝐂 = 𝐀(𝐁𝐂)

Compund linear mapping

Transpose and multiplication

(𝐀𝐁)ᵀ ≠ 𝐀ᵀ𝐁ᵀ

(𝐀𝐁)ᵀ = 𝐁ᵀ𝐀ᵀ

Square matrix multiplication

The number of rows = the number of columns

𝐀 = $[^{1\ 2} _{1\ 4}]$ ; 𝐁 = $[^{4\ -2} _{-1\ 1}]$

There is 𝐀𝐁 = 𝐁𝐀.

So these two matrices 𝐀, 𝐁 are commutative 1. 𝐀,𝐁 must be the square matrices with the same size, and the result also has the identical-size square matrix.

Power of square matrix

𝐀ᵏ = 𝐀⋅𝐀ᵏ⁻¹ = 𝐀ᵏ⁻¹⋅𝐀, where 𝐀 and 𝐀ᵏ⁻¹ are commutative because of the commutative property of matrix multiplication.

Therefore, the two positive integar powers of matrix 𝐀 are always commutative.

Power of the matrix only applicable to square matrices. Otherwise, the mismatched dimensions prevent performing matrices multiplication.

𝐀¹=𝐀 ; 𝐀⁰ = 𝐈

Thus, the range of power k is the set of natural numbers k∈ ℕ

Power of diagonal matrix

$$ 𝐀 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix} = diag\{1, 2, 3\} $$

Each elements on the main diagonal is multiplied with itself many times.

2

Scalar matrix

$$ 𝐀 = \begin{bmatrix} λ & 0 & 0 \\ 0 & λ & 0 \\ 0 & 0 & λ \end{bmatrix} = diag\{λ, λ, λ\} $$

If λ = 1, then 𝐀 = 𝐈. So 𝐈ᵏ=𝐈, and this is also applicable to zero matrix 𝟎ᵏ= 𝟎 (k≠0)

Ref

1. When are two matrices A and B: AB = BA? - StackExchange
2. 11. Powers of matrices - Massachusetts Institute of Technology