91Ó°ÊÓ

The product of the matrices with dimensions $n×m\text{and}p×q$exists if and only if $\mathbf{m}\mathbf{=}\mathbf{p}$.

Given,the matrices are:

$A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],\text{​â¶Ä‹â¶Ä‰â¶Ä‰}B=\left[\begin{array}{ccc}1& 2& 3\end{array}\right],\text{ â¶Ä‰}C=\left[\begin{array}{ccc}1& 0& −1\\ 2& 1& 0\\ 3& 2& 1\end{array}\right],\text{ â¶Ä‰}D=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\text{and}E=\left[5\right]$

The 25 matrix products obtained are:

$\begin{array}{l}AA,AB,AC,AD,AE\\ BA,BB,BC,BD,BE,\\ CA,CB,CC,CD,CE,\\ DA,DB,DC,DD,DE,\\ EA,EB,EC,ED,\text{and}EE\end{array}$

Since, the product of the matrices with dimensions$n×m\text{and}p×q$exists if and only if$m=p$. Therefore, the matrix products which exists out of above 25 products are:

$AA,BC,BD,CC,CD,DB,DE,EB,\text{ â¶Ä‰a²Ô»å}EE$

Compute the existing matrix product as:

$\begin{array}{c}AA=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\\ =\left[\begin{array}{cc}1+1& 1+1\\ 1+1& 1+1\end{array}\right]\\ =\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]\end{array}$

For BC solve as:

$\begin{array}{c}BC=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\left[\begin{array}{ccc}1& 0& −1\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]\\ =\left[\begin{array}{ccc}1+4+9& 0+2+6& −1+0+3\end{array}\right]\\ =\left[\begin{array}{ccc}14& 8& 2\end{array}\right]\end{array}$

For BD solve as:

$\begin{array}{c}BD=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\\ =[1+2+3]\\ =\left[6\right]\end{array}$

For CC solve as:

$\begin{array}{c}CC=\left[\begin{array}{ccc}1& 0& −1\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& −1\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]\\ =\left[\begin{array}{ccc}1+0−3& 0+0−2& −1+0−1\\ 2+2+0& 0+1+0& −2+0+0\\ 3+4+3& 0+2+2& −3+0+1\end{array}\right]\\ =\left[\begin{array}{ccc}−2& −2& −2\\ 4& 1& −2\\ 10& 4& −2\end{array}\right]\end{array}$

For CD solve as:

$\begin{array}{c}CD=\left[\begin{array}{ccc}1& 0& −1\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\\ =\left[\begin{array}{c}1+0−1\\ 2+1+0\\ 3+2+1\end{array}\right]\\ =\left[\begin{array}{c}0\\ 3\\ 6\end{array}\right]\end{array}$

For DB solve as:

$\begin{array}{c}DB=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\\ =\left[\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\\ 1& 2& 3\end{array}\right]\end{array}$

For DE solve as:

$\begin{array}{c}DE=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\left[5\right]\\ =\left[\begin{array}{c}5\\ 5\\ 5\end{array}\right]\end{array}$

For EB, solve as:

$\begin{array}{c}EB=\left[5\right]\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\\ =\left[\begin{array}{ccc}5& 10& 15\end{array}\right]\end{array}$

For EE solve as:

$\begin{array}{c}EE=\left[5\right]\left[5\right]\\ =\left[25\right]\end{array}$

Hence, these are the required matrix products.