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Find $\text{\hspace{0.17em}}AB-C\text{\hspace{0.17em}}$ given
On the matrix page of the calculator, we enter matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ above as the matrix variable $\text{\hspace{0.17em}}\left[A\right],$ matrix $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ above as the matrix variable $\text{\hspace{0.17em}}\left[B\right],$ and matrix $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ above as the matrix variable $\text{\hspace{0.17em}}\left[C\right].$
On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.
The calculator gives us the following matrix.
Access these online resources for additional instruction and practice with matrices and matrix operations.
Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.
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 $\text{\hspace{0.17em}}2\times 2\text{\hspace{0.17em}}$ matrix and the second is a $\text{\hspace{0.17em}}2\times 3\text{\hspace{0.17em}}$ matrix. $\text{\hspace{0.17em}}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ccc}6& 5& 4\\ 3& 2& 1\end{array}\right]\text{\hspace{0.17em}}$ has no sum.
Can we multiply any column matrix by any row matrix? Explain why or why not.
Can both the products $\text{\hspace{0.17em}}AB\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}BA\text{\hspace{0.17em}}$ be defined? If so, explain how; if not, explain why.
Yes, if the dimensions of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}m\times n\text{\hspace{0.17em}}$ and the dimensions of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}n\times m,\text{}$ both products will be defined.
Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.
Does matrix multiplication commute? That is, does $\text{\hspace{0.17em}}AB=BA?\text{\hspace{0.17em}}$ If so, prove why it does. If not, explain why it does not.
Not necessarily. To find $\text{\hspace{0.17em}}AB,\text{}$ we multiply the first row of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by the first column of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ to get the first entry of $\text{\hspace{0.17em}}AB.\text{\hspace{0.17em}}$ To find $\text{\hspace{0.17em}}BA,\text{}$ we multiply the first row of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ by the first column of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ to get the first entry of $\text{\hspace{0.17em}}BA.\text{\hspace{0.17em}}$ Thus, if those are unequal, then the matrix multiplication does not commute.
For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.
$C+D$
$\left[\begin{array}{cc}11& 19\\ 15& 94\\ 17& 67\end{array}\right]$
$B-E$
$\left[\begin{array}{cc}\mathrm{-4}& 2\\ 8& 1\end{array}\right]$
For the following exercises, use the matrices below to perform scalar multiplication.
$3B$
$\left[\begin{array}{cc}9& 27\\ 63& 36\\ 0& 192\end{array}\right]$
$\mathrm{-2}B$
$\mathrm{-4}C$
$\left[\begin{array}{cccc}\mathrm{-64}& \mathrm{-12}& \mathrm{-28}& \mathrm{-72}\\ \mathrm{-360}& \mathrm{-20}& \mathrm{-12}& \mathrm{-116}\end{array}\right]$
$\frac{1}{2}C$
$100D$
$\left[\begin{array}{ccc}1,800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& 2,100\end{array}\right]$
For the following exercises, use the matrices below to perform matrix multiplication.
$BC$
$\left[\begin{array}{cc}20& 102\\ 28& 28\end{array}\right]$
$BD$
$\left[\begin{array}{ccc}60& 41& 2\\ \mathrm{-16}& 120& \mathrm{-216}\end{array}\right]$
$CB$
$\left[\begin{array}{ccc}\mathrm{-68}& 24& 136\\ \mathrm{-54}& \mathrm{-12}& 64\\ \mathrm{-57}& 30& 128\end{array}\right]$
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.
$3D+4E$
$\left[\begin{array}{ccc}\mathrm{-8}& 41& \mathrm{-3}\\ 40& \mathrm{-15}& \mathrm{-14}\\ 4& 27& 42\end{array}\right]$
$C\mathrm{-0.5}D$
$100D\mathrm{-10}E$
$\left[\begin{array}{ccc}\mathrm{-840}& 650& \mathrm{-530}\\ 330& 360& 250\\ \mathrm{-10}& 900& 110\end{array}\right]$
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: $\text{\hspace{0.17em}}{A}^{2}=A\cdot A$ )
$BA$
$\left[\begin{array}{cc}\mathrm{-350}& 1,050\\ 350& 350\end{array}\right]$
${A}^{2}$
${B}^{2}$
$\left[\begin{array}{cc}1,400& 700\\ \mathrm{-1},400& 700\end{array}\right]$
${C}^{2}$
${B}^{2}{A}^{2}$
$\left[\begin{array}{cc}332,500& 927,500\\ \mathrm{-227},500& 87,500\end{array}\right]$
${A}^{2}{B}^{2}$
${(AB)}^{2}$
$\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right]$
${(BA)}^{2}$
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: $\text{\hspace{0.17em}}{A}^{2}=A\cdot A$ )
$AB$
$\left[\begin{array}{ccc}\mathrm{-2}& 3& 4\\ \mathrm{-7}& 9& \mathrm{-7}\end{array}\right]$
$BD$
$\left[\begin{array}{ccc}\mathrm{-4}& 29& 21\\ \mathrm{-27}& \mathrm{-3}& 1\end{array}\right]$
${D}^{2}$
$\left[\begin{array}{ccc}\mathrm{-3}& \mathrm{-2}& \mathrm{-2}\\ \mathrm{-28}& 59& 46\\ \mathrm{-4}& 16& 7\end{array}\right]$
${A}^{2}$
${D}^{3}$
$\left[\begin{array}{ccc}1& \mathrm{-18}& \mathrm{-9}\\ \mathrm{-198}& 505& 369\\ \mathrm{-72}& 126& 91\end{array}\right]$
$A(BC)$
$$\left[\begin{array}{cc}0& 1.6\\ 9& \mathrm{-1}\end{array}\right]$$
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.
$BA$
$\left[\begin{array}{ccc}2& 24& \mathrm{-4.5}\\ 12& 32& \mathrm{-9}\\ \mathrm{-8}& 64& 61\end{array}\right]$
$BC$
$\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& 10\end{array}\right]$
For the following exercises, use the matrix below to perform the indicated operation on the given matrix.
${B}^{2}$
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
${B}^{3}$
${B}^{4}$
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
${B}^{5}$
Using the above questions, find a formula for $\text{\hspace{0.17em}}{B}^{n}.\text{\hspace{0.17em}}$ Test the formula for $\text{\hspace{0.17em}}{B}^{201}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{B}^{202},\text{}$ using a calculator.
${B}^{n}=\{\begin{array}{l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\text{\hspace{1em}}n\text{\hspace{0.17em}}\text{even,}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\text{\hspace{1em}}n\text{\hspace{0.17em}}\text{odd}\text{.}\end{array}$
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