3x3 matrix: Cofactors, Determinant, Inverse, Equation solving

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Simon Thomas 6 years, 2 months ago

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Name	Status	Author	Last Modified
3x3 matrix: Cofactors, Determinant, Inverse, Equation solving	draft	Simon Thomas	07/02/2019 10:29
Simon's copy of 3x3 matrix: Cofactors, Determinant, Inverse, Equation solving	draft	Simon Thomas	01/04/2019 11:49
Simon's copy of Simon's copy of 3x3 matrix: Cofactors, Determinant, Inverse, Equation solving	draft	Simon Thomas	03/04/2019 09:25

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

cof11	integer	3
cof12	integer	-10
cof13	integer	4
cof21	integer	-2
cof22	integer	4
cof23	integer	0
cof31	integer	-3
cof32	integer	2
cof33	integer	-4

			Name	Type	Generated Value

			Name	Type	Generated Value

b1	integer	-5
b2	integer	5
b3	integer	2
vectorb	matrix	matrix([-5],[5],[2])

			Name	Type	Generated Value

matrixA	matrix	Matrix of size 3×3
a11	integer	2
a12	integer	1
a21	integer	4
a22	integer	0
a13	integer	-1
a23	integer	-3
a31	integer	2
a32	integer	1
a33	integer	1
inverseA	matrix	Matrix of size 3×3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
a22*a33-a32*a23

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Generated value: `integer`

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Parts

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Part d) Matrix entry

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Ask the student a question, and give any hints about how they should answer this part.

Calculate the nine cofactors of A= $\var{matrixA}$

(where $cof _{ij}$ represents the element in position $i,j$ of the cofactor matrix)

$cof _{11}=$ Gap 0.

$cof_{12}=$ Gap 1.

$cof_{13}=$ Gap 2.

$cof_{21}=$ Gap 3.

$cof_{22}=$ Gap 4.

$cof_{23}=$ Gap 5.

$cof_{31}=$ Gap 6.

$cof_{32}=$ Gap 7.

$cof_{33}=$ Gap 8.

Advice

$A=\left( \begin{array}{ccc} a & b & c \\d & e&f\\ g&h&j \end{array} \right),$

Cofactors are given by

Cof11 =

$+\left| \begin{array}{ccc} e&f\\ h&j \end{array} \right|,$

Cof12 =

$-\left| \begin{array}{ccc} d & f\\ g&j \end{array} \right|,$

Cof13 =

$+\left| \begin{array}{ccc} d & e\\ g&h\end{array} \right|,$

Cof21 =

$-\left| \begin{array}{ccc} b & c \\h&j \end{array} \right|,$

Cof22 =

$+\left| \begin{array}{ccc} a & c \\ g&j \end{array} \right|,$

Cof23 =

$-\left| \begin{array}{ccc} a & b \\g&h\end{array} \right|,$

Cof31 =

$+\left| \begin{array}{ccc} b & c \\e&f\end{array} \right|,$

Cof32 =

$-\left| \begin{array}{ccc} a & c \\d & f\end{array} \right|,$

Cof33 =

$+\left| \begin{array}{ccc} a & b\\d & e \end{array} \right|,$

(b) Then, the determinant, det(A) is given by the sum of the product of any row ( or column) elements by their cofactors

e.g using row 1, we obtain the determinant = $a \times cof_{11}+b \times cof_{12}+c \times cof_{13}$

$= \var{a11} \times \var{cof11} +\var{a12} \times \var{cof12}+\var{a13} \times \var{cof13} = \var{det(matrixA)}$

where the adjoint, adj(A) is given by taking the transpose of the cofactor matrix, i.e.

adj(A) =

$\left( \begin{array}{ccc} cof11 & cof21 & cof31 \\cof12 & cof22&cof32\\ cof13&cof23&cof33 \end{array} \right),$

Then $\ A^{-1}$ =

$\frac{1}{det(A)} \times \left( \begin{array}{ccc} cof11 & cof21 & cof31 \\cof12 & cof22&cof32\\ cof13&cof23&cof33 \end{array} \right),$

$\ A^{-1}$ =

$\frac{1}{\var{det(matrixA)}} \times \left( \begin{array}{ccc} \var{cof11} & \var{cof21} & \var{cof31} \\\var{cof12} & \var{cof22}&\var{cof32}\\ \var{cof13}&\var{cof23}&\var{cof33} \end{array} \right),$

$\ A^{-1}=\var{precround(inverseA,3)}$

(d) The equation $A \mathbf{x}=\mathbf{b}$ can be solved by premultiplying each side by $A^{-1}$ giving $\mathbf{x} = A^{-1}\mathbf{b}$

i.e. $\left(\matrix{x\\y\\z}\right) = \var{matrixA}^{-1} \var{vectorb}$

$\left(\matrix{x\\y\\z}\right) = \var{precround(inverseA,3)} \var{vectorb} = \var{precround(inverseA*vectorb,3)}$

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This question is used in the following exam:

Matrices Exam (Harder) by Simon Thomas in Matrices.

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3x3 matrix: Cofactors, Determinant, Inverse, Equation solving

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Simon Thomas 6 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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3x3 matrix: Cofactors, Determinant, Inverse, Equation solving

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`