11.5 Matrices and matrix operations  (Page 5/10)

No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a $2\times 2$ matrix and the second is a $2\times 3$ matrix. $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ccc}6& 5& 4\\ 3& 2& 1\end{array}\right]$ has no sum.

Yes, if the dimensions of $A$ are $m\times n$ and the dimensions of $B$ are $n\times m,$ both products will be defined.

Not necessarily. To find $AB,$ we multiply the first row of $A$ by the first column of $B$ to get the first entry of $AB.$ To find $BA,$ we multiply the first row of $B$ by the first column of $A$ to get the first entry of $BA.$ Thus, if those are unequal, then the matrix multiplication does not commute.

$\left[\begin{array}{cc}11& 19\\ 15& 94\\ 17& 67\end{array}\right]$

$\left[\begin{array}{cc}\mathrm{-4}& 2\\ 8& 1\end{array}\right]$

Undidentified; dimensions do not match

$\left[\begin{array}{cc}9& 27\\ 63& 36\\ 0& 192\end{array}\right]$

$\left[\begin{array}{cccc}\mathrm{-64}& \mathrm{-12}& \mathrm{-28}& \mathrm{-72}\\ \mathrm{-360}& \mathrm{-20}& \mathrm{-12}& \mathrm{-116}\end{array}\right]$

$\left[\begin{array}{ccc}1,800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& 2,100\end{array}\right]$

$\left[\begin{array}{cc}20& 102\\ 28& 28\end{array}\right]$

$\left[\begin{array}{ccc}60& 41& 2\\ \mathrm{-16}& 120& \mathrm{-216}\end{array}\right]$

$\left[\begin{array}{ccc}\mathrm{-68}& 24& 136\\ \mathrm{-54}& \mathrm{-12}& 64\\ \mathrm{-57}& 30& 128\end{array}\right]$

Undefined; dimensions do not match.

$\left[\begin{array}{ccc}\mathrm{-8}& 41& \mathrm{-3}\\ 40& \mathrm{-15}& \mathrm{-14}\\ 4& 27& 42\end{array}\right]$

$\left[\begin{array}{ccc}\mathrm{-840}& 650& \mathrm{-530}\\ 330& 360& 250\\ \mathrm{-10}& 900& 110\end{array}\right]$

$\left[\begin{array}{cc}\mathrm{-350}& 1,050\\ 350& 350\end{array}\right]$

Undefined; inner dimensions do not match.

$\left[\begin{array}{cc}1,400& 700\\ \mathrm{-1},400& 700\end{array}\right]$

$\left[\begin{array}{cc}332,500& 927,500\\ \mathrm{-227},500& 87,500\end{array}\right]$

$\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right]$

$\left[\begin{array}{ccc}\mathrm{-2}& 3& 4\\ \mathrm{-7}& 9& \mathrm{-7}\end{array}\right]$

$\left[\begin{array}{ccc}\mathrm{-4}& 29& 21\\ \mathrm{-27}& \mathrm{-3}& 1\end{array}\right]$

$\left[\begin{array}{ccc}\mathrm{-3}& \mathrm{-2}& \mathrm{-2}\\ \mathrm{-28}& 59& 46\\ \mathrm{-4}& 16& 7\end{array}\right]$

$\left[\begin{array}{ccc}1& \mathrm{-18}& \mathrm{-9}\\ \mathrm{-198}& 505& 369\\ \mathrm{-72}& 126& 91\end{array}\right]$

$$\left[\begin{array}{cc}0& 1.6\\ 9& \mathrm{-1}\end{array}\right]$$

$\left[\begin{array}{ccc}2& 24& \mathrm{-4.5}\\ 12& 32& \mathrm{-9}\\ \mathrm{-8}& 64& 61\end{array}\right]$

$\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& 10\end{array}\right]$

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$B^n=\{\begin{array}{l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],n\text{even,}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],n\text{odd}\text{.}\end{array}$