Determinants

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Christian Lawson-Perfect

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Christian Lawson-Perfect 10 years ago

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Christian Lawson-Perfect 10 years ago

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Hayley Bishop 11 years, 8 months ago

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Question statement

Let

$A=\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix},\;\; B=\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix},\;\; C=\begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a21	integer	-3
a22	integer	6
b22	integer	-4
b21	integer	-2
b1	integer	5
s2	integer	-1
s1	integer	-1
b12	integer	2
b11	integer	8
tr1	integer	6
c12	integer	6
c11	integer	4
tr2	integer	8
tr4	integer	9
c22	integer	9
a11	integer	-1
a12	integer	4
db	integer	-28
dc	integer	6
da	integer	6
a1	integer	2
c21	integer	5
c1	integer	6
tr3	integer	-4
a	integer	1
b	integer	-6
s	integer	-1
u	integer	1
t	integer	1

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Value

xxxxxxxxxx
 
s2*random(1..6)

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Generated value: `integer`

-3

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Parts

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1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number entry
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Calculate the determinants of these matrices:

$\mathrm{det}\left(A\right) = \;\;$ Gap 0. $,\;\;\;\mathrm{det}\left(B\right) = \;\;$ Gap 1. $,\;\;\;\mathrm{det}\left(C\right) = \;\;$ Gap 2. $,\;\;\;\mathrm{det}\left(ABC\right) = \;\;$ Gap 3.

Advice

a)

Determinants.

Here is the formula for the determinant of a $2 \times 2$ matrix:

$M = \begin{pmatrix} a & b \\ c&d \end{pmatrix} \Rightarrow \mathrm{det}\left(M\right) = ad-bc$

$\mathrm{det}\left(A\right) = \simplify[]{{a11}*{a22}-{a12}*{a21} = {dA}}$
$\mathrm{det}\left(B\right) = \simplify[]{{b11}*{b22}-{b12}*{b21} = {dB}}$
$\mathrm{det}\left(C\right) = \simplify[]{{c11}*{c22}-{c12}*{c21} = {dC}}$

Determinant of a product of matrices.

If we have two $n \times n$ matrices $M$ and $N$ then:

$\mathrm{det}\left(MN\right) = \mathrm{det}\left(M\right)\mathrm{det}\left(N\right)$
And it follows that if we have a third matrix

$P$ that:

$\mathrm{det}\left(MNP\right) = \mathrm{det}\left(M\right)\mathrm{det}\left(N\right)\mathrm{det}\left(P\right)$

Thus for our example we have:

$\begin{eqnarray*}\mathrm{det}\left(ABC\right) &=& \mathrm{det}\left(A\right)\times\mathrm{det}\left(B\right)\times\mathrm{det}\left(C\right)\\ &=& \var{dA}\times \var{dB} \times \var{dC}\\ &=& \var{dA*dB*dC} \end{eqnarray*}$

Inverse of a $2 \times 2$ matrix

Suppose $M$ is a $2 \times 2$ matrix and $\mathrm{det}\left(M\right) = \Delta \neq 0$ .

Then $M$ is invertible and:

$M = \begin{pmatrix} a & b \\ c&d \end{pmatrix} \Rightarrow M^{-1} = \frac{1}{\Delta} \begin{pmatrix} d & -b\\ -c& a \end{pmatrix}=\begin{pmatrix} \frac{d}{\Delta} & -\frac{b}{\Delta}\\ -\frac{c}{\Delta}& \frac{a}{\Delta} \end{pmatrix}$

Applying this to these examples we obtain:

b)

$A^{-1} = \begin{pmatrix} \simplify[std]{{a22}/{dA}} &\simplify[std]{{-a12}/{dA}}\\\simplify[std]{{-a21}/{dA}}&\simplify[std]{{a11}/{dA}}\end{pmatrix}$

c)

$B^{-1} = \begin{pmatrix} \simplify[std]{{b22}/{dB}} &\simplify[std]{{-b12}/{dB}}\\\simplify[std]{{-b21}/{dB}}&\simplify[std]{{b11}/{dB}}\end{pmatrix}$

d)

$C^{-1} = \begin{pmatrix} \simplify[std]{{c22}/{dC}} &\simplify[std]{{-c12}/{dC}}\\\simplify[std]{{-c21}/{dC}}&\simplify[std]{{c11}/{dC}}\end{pmatrix}$

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This question is used in the following exams:

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Checkpoint description

Christian Lawson-Perfect 10 years ago

Christian Lawson-Perfect 10 years ago

Hayley Bishop 11 years, 8 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

Gap-fill

a)

Determinants.

Determinant of a product of matrices.

Inverse of a $2 \times 2$ matrix

b)

c)

d)

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Determinants

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 10 years ago

Christian Lawson-Perfect 10 years ago

Hayley Bishop 11 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

Determinants.

Determinant of a product of matrices.

Inverse of a 2×22 \times 2 matrix

b)

c)

d)

Generated value: `integer`

Inverse of a $2 \times 2$ matrix