Here are some standard properties of matrix multiplication: Let [math]A[/math] be an m x n matrix, and let [math]B[/math] and [math]C[/math] have sizes for which the indicated sums and products are defined.[br][br][list=1][*][math]A(BC)=(AB)C[/math][br][/*][*][math]A(B+C)=AB+AC[/math][br][/*][*][math](B+C)A=BA+CA[/math][br][/*][*][math]r(AB)=(rA)B=A(rB)[/math] for any real number [math]r[/math][/*][*][math]I_mA=A=AI_{_n}[/math][br][/*][/list][br]We also have some useful properties regarding inverses: Let [math]A[/math] and [math]B[/math] be invertible matrices of the same size.[br][br][list=1][*][math]AB[/math] is invertible and [math]\left(AB\right)^{-1}=B^{-1}A^{-1}[/math][/*][*][math]\left(rA\right)^{-1}=\frac{1}{r}A^{-1}[/math] for any non-zero real number [math]r[/math][/*][*][math]\left(A^{-1}\right)^{-1}=A[/math][br][/*][/list]

Given an n x n matrix [math]A[/math] and a positive integer [math]k[/math], [math]A^k[/math] denotes the product of [math]k[/math] copies of [math]A[/math]: [math]A^k=\underbrace{A\cdots A}_{k}[/math]. By convention, [math]A^0=I[/math].[br][br]From the first property of the inverses, we can see that [math]\left(A^k\right)^{-1}=\left(A^{-1}\right)^k[/math]. Sometimes it is abbreviated as [math]A^{-k}[/math].[br][br][u]Caution[/u]: In general, [math]\left(AB\right)^k\ne A^kB^k[/math] because [math](AB)^k=\underbrace{(AB)(AB)\cdots(AB)}_{k}[/math]. And it equals [math]A^kB^k[/math] only when [math]AB=BA[/math].[br][br]

Given an m x n matrix [math]A[/math], the [b]transpose[/b] of [math]A[/math] is the n x m matrix, denoted by [math]A^T[/math], whose columns are formed from the corresponding rows of [math]A[/math] i.e the [math]i^{\text{th}}[/math] column of [math]A^T[/math] is the [math]i^{\text{th}}[/math] row of [math]A[/math].[br][br][u]Examples[/u]: [br][math]A=\begin{pmatrix} 1 & 2 \\ 3 & 4\end{pmatrix}, A^T=\begin{pmatrix} 1 & 3 \\2 & 4\end{pmatrix}[/math][br][math]B=\begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\end{pmatrix}, B^T=\begin{pmatrix} 1 & 4 \\2 & 5 \\ 3 & 6\end{pmatrix}[/math][br][math]C=\begin{pmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ 7 & 8\end{pmatrix}, C^T=\begin{pmatrix} 1 & 3 & 5 & 7 \\2 & 4 & 6 & 8\end{pmatrix}[/math][br][br]The following are some useful properties regarding transpose: Let A and B denote matrices having sizes for which the indicated sums and products are defined.[br][br][list=1][*][math]\left(A^T\right)^T=A[/math][/*][*] [math]\left(A+B\right)^T=A^T+B^T[/math][/*][*][math]\left(rA\right)^T=rA^T[/math] for any real number [math]r[/math][/*][*][math]\left(AB\right)^T=B^TA^T[/math][br][/*][/list][br][br][br]The following are some questions about matrix algebra: