Matrices

Multiplication properties

Property 1: Associative Property of Multiplication

Given A_mxn , B_nxp and C_pxq , then A·(B·C)=(A·B)·C .

$\fs2Given\;A=\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right),\;B=\left(\begin{matrix}3&-1\\1&0\end{matrix}\right)\;and\;C=\left(\begin{matrix}1&2\\-1&0\end{matrix}\right)$
$\fs2A\cdot\;(B\cdot\;C)=\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)\cdot\left[\left(\begin{matrix}3&-1\\1&0\end{matrix}\right)\cdot\left(\begin{matrix}1&2\\-1&0\end{matrix}\right)\\right]=\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)\cdot\left(\begin{matrix}4&6\\1&2\end{matrix}\right)=\left(\begin{matrix}7&10\\-3&-6\end{matrix}\right)$
$\fs2(A\cdot\;B)\cdot\;C=\left[\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)\cdot\left(\begin{matrix}3&-1\\1&0\end{matrix}\right)\\right]\cdot\left(\begin{matrix}1&2\\-1&0\end{matrix}\right)=\left(\begin{matrix}5&-2\\-3&0\end{matrix}\right)\cdot\left(\begin{matrix}1&2\\-1&0\end{matrix}\right)=\left(\begin{matrix}7&10\\-3&-6\end{matrix}\right)$

Property 2: Multiplication of Matrices is not Commutative
Given A_mxn and B_nxm, then A·B ≠ B·A.
(AB does not have to = BA)

$\fs2Given\;A=\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)\;and\;B=\left(\begin{matrix}3&-1\\1&0\end{matrix}\right)$
$\fs2A\cdot\;B=\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)\cdot\left(\begin{matrix}3&-1\\1&0\end{matrix}\right)=\left(\begin{matrix}5&-2\\-3&0\end{matrix}\right)$
$\fs2B\cdot\;A=\left(\begin{matrix}3&-1\\1&0\end{matrix}\right)\cdot\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)=\left(\begin{matrix}6&0\\2&-1\end{matrix}\right)$

Property 3: Distributive Property of Multiplication
Given A_mxn , B_nxp and C_nxp , then A·(B+C)=A·B +A·C
Given A_pxq , B_nxp and C_nxp,then (B+C)·A=B·A +C·A

Property 4: Multiplicative identity

Given A_mxn, There are unique matrices I_n and I_m with A_mxn·I_n=A_mxn and I_m·A_mxn=A_mxn

The matrix I_n, called the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere.

$\fs2A\cdot\;I=\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)=\left(\begin{matrix}2&-1\\0&-3\end{matrix}\right)$

Property 5: Null Matrices may have Non-null Divisors.

The matrix product AB can be zero although A ≠ 0 and B ≠0.
Similar, it is possible that A ≠0, A² ≠ 0, . . . , but A^p = 0.

$\fs2Given\;A=\left(\begin{matrix}1&1\\-2&-2\end{matrix}\right)\;and\;\;B=\left(\begin{matrix}3&-1\\-3&1\end{matrix}\right)$
$\fs2then\;A\cdot\;B=0\;;\;\;\left(\begin{matrix}1&1\\-2&-2\end{matrix}\right)\cdot\left(\begin{matrix}3&-1\\-3&1\end{matrix}\right)=\left(\begin{matrix}0&0\\0&0\end{matrix}\right)$

No se verifica la ley de cancelación