# 7.5 Matrices and matrix operations  (Page 5/10)

 Page 5 / 10

## Using a calculator to perform matrix operations

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.

$\left[A\right]×\left[B\right]-\left[C\right]$

The calculator gives us the following matrix.

$\left[\begin{array}{rrr}\hfill -983& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-462& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}136\\ \hfill 1,820& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}1,897& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}-856\\ \hfill -311& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}2,032& \hfill \text{\hspace{0.17em}}\text{\hspace{0.17em}}413\end{array}\right]$

Access these online resources for additional instruction and practice with matrices and matrix operations.

## Key concepts

• A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
• The dimensions of a matrix refer to the number of rows and the number of columns. A $\text{\hspace{0.17em}}3×2\text{\hspace{0.17em}}$ matrix has three rows and two columns. See [link] .
• Scalar multiplication involves multiplying each entry in a matrix by a constant. See [link] .
• Scalar multiplication is often required before addition or subtraction can occur. See [link] .
• Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
• The product of two matrices, $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B,$ is obtained by multiplying each entry in row 1 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by each entry in column 1 of $\text{\hspace{0.17em}}B;\text{\hspace{0.17em}}$ then multiply each entry of row 1 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by each entry in columns 2 of $\text{\hspace{0.17em}}B,\text{}$ and so on. See [link] and [link] .
• Many real-world problems can often be solved using matrices. See [link] .
• We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See [link] .

## Verbal

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 $\text{\hspace{0.17em}}2×2\text{\hspace{0.17em}}$ matrix and the second is a $\text{\hspace{0.17em}}2×3\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}m×n\text{\hspace{0.17em}}$ and the dimensions of $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}n×m,\text{}$ 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.

## Algebraic

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

$A=\left[\begin{array}{cc}1& 3\\ 0& 7\end{array}\right],B=\left[\begin{array}{cc}2& 14\\ 22& 6\end{array}\right],C=\left[\begin{array}{cc}1& 5\\ 8& 92\\ 12& 6\end{array}\right],D=\left[\begin{array}{cc}10& 14\\ 7& 2\\ 5& 61\end{array}\right],E=\left[\begin{array}{cc}6& 12\\ 14& 5\end{array}\right],F=\left[\begin{array}{cc}0& 9\\ 78& 17\\ 15& 4\end{array}\right]$

$A+B$

$C+D$

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

$A+C$

$B-E$

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

$C+F$

$D-B$

Undidentified; dimensions do not match

For the following exercises, use the matrices below to perform scalar multiplication.

$A=\left[\begin{array}{rr}\hfill 4& \hfill 6\\ \hfill 13& \hfill 12\end{array}\right],B=\left[\begin{array}{rr}\hfill 3& \hfill 9\\ \hfill 21& \hfill 12\\ \hfill 0& \hfill 64\end{array}\right],C=\left[\begin{array}{rrrr}\hfill 16& \hfill 3& \hfill 7& \hfill 18\\ \hfill 90& \hfill 5& \hfill 3& \hfill 29\end{array}\right],D=\left[\begin{array}{rrr}\hfill 18& \hfill 12& \hfill 13\\ \hfill 8& \hfill 14& \hfill 6\\ \hfill 7& \hfill 4& \hfill 21\end{array}\right]$

$5A$

$3B$

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

$-2B$

$-4C$

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

$\frac{1}{2}C$

$100D$

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

For the following exercises, use the matrices below to perform matrix multiplication.

$A=\left[\begin{array}{rr}\hfill -1& \hfill 5\\ \hfill 3& \hfill 2\end{array}\right],B=\left[\begin{array}{rrr}\hfill 3& \hfill 6& \hfill 4\\ \hfill -8& \hfill 0& \hfill 12\end{array}\right],C=\left[\begin{array}{rr}\hfill 4& \hfill 10\\ \hfill -2& \hfill 6\\ \hfill 5& \hfill 9\end{array}\right],D=\left[\begin{array}{rrr}\hfill 2& \hfill -3& \hfill 12\\ \hfill 9& \hfill 3& \hfill 1\\ \hfill 0& \hfill 8& \hfill -10\end{array}\right]$

$AB$

$BC$

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

$CA$

$BD$

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

$DC$

$CB$

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

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.

$A=\left[\begin{array}{rr}\hfill 2& \hfill -5\\ \hfill 6& \hfill 7\end{array}\right],B=\left[\begin{array}{rr}\hfill -9& \hfill 6\\ \hfill -4& \hfill 2\end{array}\right],C=\left[\begin{array}{rr}\hfill 0& \hfill 9\\ \hfill 7& \hfill 1\end{array}\right],D=\left[\begin{array}{rrr}\hfill -8& \hfill 7& \hfill -5\\ \hfill 4& \hfill 3& \hfill 2\\ \hfill 0& \hfill 9& \hfill 2\end{array}\right],E=\left[\begin{array}{rrr}\hfill 4& \hfill 5& \hfill 3\\ \hfill 7& \hfill -6& \hfill -5\\ \hfill 1& \hfill 0& \hfill 9\end{array}\right]$

$A+B-C$

$4A+5D$

Undefined; dimensions do not match.

$2C+B$

$3D+4E$

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

$C-0.5D$

$100D-10E$

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

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: $\text{\hspace{0.17em}}{A}^{2}=A\cdot A$ )

$A=\left[\begin{array}{rr}\hfill -10& \hfill 20\\ \hfill 5& \hfill 25\end{array}\right],B=\left[\begin{array}{rr}\hfill 40& \hfill 10\\ \hfill -20& \hfill 30\end{array}\right],C=\left[\begin{array}{rr}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right]$

$AB$

$BA$

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

$CA$

$BC$

Undefined; inner dimensions do not match.

${A}^{2}$

${B}^{2}$

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

${C}^{2}$

${B}^{2}{A}^{2}$

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

${A}^{2}{B}^{2}$

${\left(AB\right)}^{2}$

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

${\left(BA\right)}^{2}$

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: $\text{\hspace{0.17em}}{A}^{2}=A\cdot A$ )

$A=\left[\begin{array}{rr}\hfill 1& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right],B=\left[\begin{array}{rrr}\hfill -2& \hfill 3& \hfill 4\\ \hfill -1& \hfill 1& \hfill -5\end{array}\right],C=\left[\begin{array}{rr}\hfill 0.5& \hfill 0.1\\ \hfill 1& \hfill 0.2\\ \hfill -0.5& \hfill 0.3\end{array}\right],D=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill -1\\ \hfill -6& \hfill 7& \hfill 5\\ \hfill 4& \hfill 2& \hfill 1\end{array}\right]$

$AB$

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

$BA$

$BD$

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

$DC$

${D}^{2}$

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

${A}^{2}$

${D}^{3}$

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

$\left(AB\right)C$

$A\left(BC\right)$

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

## Technology

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.

$A=\left[\begin{array}{rrr}\hfill -2& \hfill 0& \hfill 9\\ \hfill 1& \hfill 8& \hfill -3\\ \hfill 0.5& \hfill 4& \hfill 5\end{array}\right],B=\left[\begin{array}{rrr}\hfill 0.5& \hfill 3& \hfill 0\\ \hfill -4& \hfill 1& \hfill 6\\ \hfill 8& \hfill 7& \hfill 2\end{array}\right],C=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 1& \hfill 0& \hfill 1\end{array}\right]$

$AB$

$BA$

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

$CA$

$BC$

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

$ABC$

## Extensions

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

$B=\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0\end{array}\right]$

${B}^{2}$

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

${B}^{3}$

${B}^{4}$

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

${B}^{5}$

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.

${B}^{n}=\left\{\begin{array}{l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\text{ }n\text{\hspace{0.17em}}\text{even,}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\text{ }n\text{\hspace{0.17em}}\text{odd}\text{.}\end{array}$

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