Matrices commuting means that two matrices $A$ and $B$ can be multiplied in any order, and the result will be the same. In other words, matrices $A$ and $B$ commute if:

$AB = BA$

Explanation:

1. Matrix Multiplication: Normally, matrix multiplication is not commutative, meaning $AB \neq BA$ in general.

2. When They Commute: If $AB = BA$, then we say that $A$ and $B$ commute. This is a special condition that holds only for certain pairs of matrices.

Example:

Consider two matrices:

$A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$

If you compute $AB$ and $BA$:

$AB = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix}$

$BA = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}$

Since $AB \neq BA$, these matrices do not commute.

Special Cases:

1. Diagonal Matrices: Diagonal matrices commute with each other.
2. Identity Matrix: Any matrix commutes with the identity matrix, since $AI = IA = A$.
3. Scalar Matrices: Scalar multiples of matrices often commute.

Would you like more details on conditions when matrices commute or further examples?

Related Questions:

1. What are examples of matrices that always commute?
2. How can we prove whether two given matrices commute?
3. Can non-square matrices ever commute?
4. Why is matrix multiplication not commutative in general?
5. What are some applications of commuting matrices in linear algebra?

Tip: If you're studying commutating matrices, focus on cases like diagonal matrices or operators in quantum mechanics, where commuting properties are crucial.