Maths Support: Matrix arithmetic

Question 1

Let
\[A=\simplify{{a}},\;\; B=\simplify{{b}},\;\; C=\simplify{{c}}\]
Calculate the following $2 \times 2$ matrices:

a)

$\mathrm{A}+\mathrm{B} = \simplify[std]{{a}+{b}} = $

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b)

$\simplify{{p}A+{q}B = {p}{a}+{q}{b}}=$

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c)

$\simplify{{p1}A+{q1}B+{r1}C = {p1}{a}+{q1}{b}+{r1}{c}}=$

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a)

\[ \begin{eqnarray*} \simplify[std]{A+B} &=&\simplify[std]{{a}+{b}}\\ &=& \begin{pmatrix} \simplify[std]{{a[0][0]}+{b[0][0]}}& \simplify[std]{{a[0][1]}+{b[0][1]}}\\ \simplify[std]{{a[1][0]}+{b[1][0]}}&\simplify[std]{{a[1][1]}+{b[1][1]}} \end{pmatrix}\\ &=&\simplify{{apb}}\\ \end{eqnarray*} \]

b)

\[ \begin{eqnarray*} \simplify[std]{{p}A+{q}B} &=&\simplify[std]{{p}{a}+{q}{b}}\\ &=& \begin{pmatrix} \simplify[std]{{p}*{a[0][0]}+{q}*{b[0][0]}}& \simplify[std]{{p}*{a[0][1]}+{q}*{b[0][1]}}\\ \simplify[std]{{p}*{a[1][0]}+{q}*{b[1][0]}}&\simplify[std]{{p}*{a[1][1]}+{q}*{b[1][1]}} \end{pmatrix}\\ &=&\simplify{{lcab}}\\ \end{eqnarray*} \]

c)

\[ \begin{eqnarray*} \simplify[std]{{p1}A+{q1}B+{r1}C} &=&\simplify[std]{{p1}{a}+{q1}{b}+{r1}{c}}\\ &=& \begin{pmatrix} \simplify[std]{{p1}*{a[0][0]}+{q1}*{b[0][0]}+{r1}*{c[0][0]}}& \simplify[std]{{p1}*{a[0][1]}+{q1}*{b[0][1]}+{r1}*{c[0][1]}}\\ \simplify[std]{{p1}*{a[1][0]}+{q1}*{b[1][0]}+{r1}*{c[1][0]}}&\simplify[std]{{p1}*{a[1][1]}+{q1}*{b[1][1]}+{r1}*{c[1][1]}} \end{pmatrix}\\ &=&\simplify{{lcabc}}\\ \end{eqnarray*} \]

Question 2

Do the following matrix problems
Let
\[A=\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix},\;\; B=\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix},\;\; C=\begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}\]
Calculate the following products of these matrices:

a)

$AB = \begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix} = $

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b)

$BA = \begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}=$

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c)

$CB = \begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix} \begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}=$

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d)

$AC = \begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}=$

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a)

\[ \begin{eqnarray*} AB &=& \begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{a11}{b11}+{a12}{b21}}&\simplify[]{{a11}{b12}+{a12}{b22}}\\ \simplify[]{{a21}{b11}+{a22}{b21}}&\simplify[]{{a21}{b12}+{a22}{b22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{ab11}&\var{ab12}\\ \var{ab21}&\var{ab22}\\ \end{pmatrix} \end{eqnarray*} \]

b)

\[ \begin{eqnarray*} BA &=& \begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{b11}{a11}+{b12}{a21}}&\simplify[]{{b11}{a12}+{b12}{a22}}\\ \simplify[]{{b21}{a11}+{b22}{a21}}&\simplify[]{{b21}{a12}+{b22}{a22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{ba11}&\var{ba12}\\ \var{ba21}&\var{ba22}\\ \end{pmatrix} \end{eqnarray*} \]

c)

\[ \begin{eqnarray*} CB &=& \begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{c11}{b11}+{c12}{b21}}&\simplify[]{{c11}{b12}+{c12}{b22}}\\ \simplify[]{{c21}{b11}+{c22}{b21}}&\simplify[]{{c21}{b12}+{a22}{b22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{cb11}&\var{cb12}\\ \var{cb21}&\var{cb22}\\ \end{pmatrix} \end{eqnarray*} \]

d)

\[ \begin{eqnarray*} AC &=& \begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}\begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \simplify[]{{a11}{c11}+{a12}{c21}}&\simplify[]{{a11}{c12}+{a12}{c22}}\\ \simplify[]{{a21}{c11}+{a22}{c21}}&\simplify[]{{a21}{c12}+{a22}{c22}}\\ \end{pmatrix}\\ &=& \begin{pmatrix} \var{ac11}&\var{ac12}\\ \var{ac21}&\var{ac22}\\ \end{pmatrix} \end{eqnarray*} \]

Question 3

Answer the following questions on matrices.

a)

Let

\[A = \left(\begin{array}{rrr} \var{a11} & \var{a12} & \var{a13}\\ \var{a21} & \var{a22} & \var{a23}\\ \var{a31} & \var{a32} & \var{a33}\\ \end{array}\right),\;\;\;\; B= \left(\begin{array}{rrr} \var{b11} & \var{b12} & \var{b13}\\ \var{b21} & \var{b22} & \var{b23}\\ \var{b31} & \var{b32} & \var{b33}\\ \end{array}\right),\;\;\;\; v= \left(\begin{array}{r} \var{v1}\\ \var{v2} \\ \var{v3} \end{array}\right),\;\;\;\; w= \left(\begin{array}{r} \var{w1}\\ \var{w2} \\ \var{w3} \end{array}\right)\]

Find the following products:

$Av = $

Expected answer:

$Bw = $

Expected answer:

$BA = $

Expected answer:

$AB = $

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b)

Consider the following matrices together with the matrices from the first part of the question.

\[\begin{eqnarray}&C=& \var{mac},\;\;\;\; &D=& \var{mad},\;\;\; \;&E= &\var{mae}\\&F=& \left(\begin{array}{rr} \var{w1} & \var{a12}\\ \var{w2} & \var{b23} \\ \var{w3} & \var{w2} \\\var{v1} & \var{b12}\\ 0 & \var{-w2} \end{array}\right),\;\;\;\;&G=&\var{mag},\;\;\;\;&H=&\var{mah} \end{eqnarray}\]

Which of the following products of matrices can be calculated?

Gap 0

	Can be calculated	Cannot be calculated
$BC$
$EF$
$AG$
$CD$
$GH$
$AE$
$GB$
$FE$
$HE$
$DC$

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Numbas has failed

Maths Support: Matrix arithmetic

Question 1

Question 1

a)

b)

c)

Advice

a)

b)

c)

Question 2

Question 2

a)

b)

c)

d)

Advice

a)

b)

c)

d)

Question 3

Question 3

a)

b)

Gap 0

Advice