Matrices follow specific algebraic rules which differ in many ways from regular arithmetic. Understanding these properties helps simplify complex expressions and solve matrix-related problems effectively.
\[ AB \neq BA \]
Matrix multiplication is not commutative. The order of multiplication matters.
\[ A(BC) = (AB)C \]
Matrix multiplication is associative. You can group matrices when multiplying.
\[ A(B + C) = AB + AC \]
\[ (A + B)C = AC + BC \]
These are distributive properties of matrix multiplication over addition.
\[ (AB)^T = B^T A^T \]
The transpose of a product is the product of the transposes in reverse order.
\[ |AB| = |A||B| \]
The determinant of a product equals the product of the determinants.
\[ A^n = AA \cdots A \]
Raising a matrix to the power \( n \) means multiplying it by itself \( n \) times.
\[ A^0 = I \]
Any square matrix raised to the power 0 gives the identity matrix.
\[ (\alpha A)^n = \alpha^n A^n \]
Scalar multiplication can be distributed across matrix powers.
\[ |A^n| = |A|^n \]
The determinant of a matrix power equals the determinant raised to that power.
\[ I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} \]
The identity matrix acts as the multiplicative identity in matrix algebra.