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Tamsin Smith 7 months, 1 week ago

Published this.

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Tamsin Smith 9 months, 3 weeks ago

Created this as a copy of W1b - Cofactors, Determinant and Inverse of a 3x3 matrix.

Name	Status	Author	Last Modified
Multiply two matrices	Ready to use	Christian Lawson-Perfect	01/06/2016 11:06
Lois's copy of Multiply two matrices	draft	Lois Rollings	26/08/2016 11:46
Simon's copy of Lois's copy of Multiply two matrices	draft	Simon Vaughan	25/01/2017 14:35
Francis 's copy of Simon's copy of Lois's copy of Multiply two matrices	draft	Francis Duah	22/11/2016 23:11
Francis 's copy of Francis 's copy of Simon's copy of Lois's copy of Multiply two matrices	draft	Francis Duah	22/11/2016 23:13
EG3101 EG5116 EG4103 Matrices: Multiplying two 2x2 matrices	draft	bryan pearce	12/01/2017 14:38
EG3101 EG5116 EG4103 Matrices: Multiplying two 3x3 matrices	draft	bryan pearce	12/01/2017 15:11
EG3101 EG5116 EG4103 Matrices: adding and subtracting two 3x3 matrices	draft	bryan pearce	18/10/2017 17:58
EG3101 EG5116 EG4103 Matrices: determinant and inverse of a 2x2 matrix	draft	bryan pearce	12/01/2017 16:53
EG3101 EG5116 EG4103 Matrices: cofactors determinant and inverse of a 3x3 matrix	draft	bryan pearce	17/01/2017 12:38
EG5116 EG4103 Matrices: Cramers Rule applied to 3 simultaneous equations	draft	bryan pearce	20/01/2017 17:56
Denis's copy of EG5116 EG4103 Matrices: Cramers Rule applied to 3 simultaneous equations	draft	Denis Flynn	23/03/2020 16:46
Alex's copy of Simon's copy of Lois's copy of Multiply two matrices	draft	Alex Van den Hof	26/01/2017 14:17
Owen's copy of Multiply two matrices	draft	Owen Jepps	31/03/2020 08:21
Andrew's copy of Matrices: Cramers Rule 3x3	draft	Andrew Dunbar	30/06/2017 20:34
Andrew's copy of Matrices: Cramers Rule 3x3	draft	Andrew Dunbar	30/06/2017 22:04
Ex Sheet 6 Cofactors, Determinant and inverse of a 3x3 matrix	draft	Violeta CIT	21/09/2017 19:19
Ex 6 Cofactors, Determinant and Inverse of a 3x3 matrix	Ready to use	Violeta CIT	04/10/2017 16:45
Question 5 MATH6005 Assessment 1 Cofactors, Determinant and Inverse of a 3x3 matrix	Ready to use	Violeta CIT	04/10/2017 17:56
Week 13: Matrices: Cramers Rule 3x3	draft	Ashley Cusack	12/12/2017 14:37
Pelle's copy of Ashley's copy of Andrew's copy of Matrices: Cramers Rule 3x3	Ready to use	Pelle Englund	10/12/2021 19:02
Marisa's copy of Week 13: Matrices: Cramers Rule 3x3	draft	Marisa Llorens	17/01/2018 12:42
Adrian's copy Matrices: adding and subtracting two 3x3 matrices	draft	Adrian Jannetta	16/08/2023 14:56
Matrices: adding and subtracting two 3x3 matrices	draft	Marie Nicholson	04/02/2021 15:44
Ex 6 Cofactors, Determinant and Inverse of a 3x3 matrix	Ready to use	Violeta CIT	16/09/2018 12:34
Cofactors, Determinant and Inverse of a 3x3 matrix (Adaptive)	Ready to use	Clodagh Carroll	27/09/2019 13:37
Custom Marking version of Ex 6 Cofactors, Determinant and Inverse of a 3x3 matrix (CC)	Doesn't work	Clodagh Carroll	27/09/2018 17:08
William's copy of Multiply two matrices	draft	William Lee	18/12/2018 13:00
Alison's copy of Matrices: cofactors determinant and inverse of a 3x3 matrix	draft	Alison Loddick	29/01/2019 08:38
Lois's copy of Multiply two matrices	draft	Lois Rollings	05/06/2019 16:06
Cofactors, Determinant and Inverse of a 3x3 matrix	draft	Xiaodan Leng	10/07/2019 23:30
Paula's copy of Adaptive Marking version of Ex 6 Cofactors, Determinant and Inverse of a 3x3 matrix (CC)	draft	Paula Ruiz-Hincapie	04/09/2019 11:14
Matt's copy of Ex 6 Cofactors, Determinant and Inverse of a 3x3 matrix	draft	Matt Tranter	16/09/2019 12:29
W1b - Cofactors, Determinant and Inverse of a 3x3 matrix	Ready to use	Timur Zaripov	17/06/2022 09:37
Cofactors, Determinant and Inverse of a 3x3 matrix (Adaptive) draft 2	draft	Clodagh Carroll	25/09/2020 14:45
Cramers Rule 3x3	draft	Shaheen Charlwood	19/02/2020 21:34
Andrew's copy of Multiply two matrices	Ready to use	Andrew Hogue	22/06/2020 17:12
Inverse of a 3x3 matrix	draft	Tamsin Smith	17/09/2024 13:21

There are 73 other versions that do you not have access to.

Question statement

Inverse of a $3 \times 3$ Matrix

Follow the steps in the questions to find the inverse of a $3 \times 3$ matrix

Give any introductory information the student needs.

			Name	Type	Generated Value

cof11	integer	40
cof12	integer	-23
cof13	integer	-2
cof21	integer	-74
cof22	integer	-57
cof23	integer	58
cof31	integer	-42
cof32	integer	-3
cof33	integer	-16

			Name	Type	Generated Value

			Name	Type	Generated Value

matrixA	matrix	Matrix of size 3×3
a11	integer	-3
a12	integer	10
a21	integer	1
a22	integer	2
a13	integer	6
a23	integer	-3
a31	integer	4
a32	integer	6
a33	integer	11
inverseA	matrix	Matrix of size 3×3
detA	integer	-362

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
a22*a33-a32*a23

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Description

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Generated value: `integer`

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Let:

$A=\var{matrixA}$

Find the determinant of $A$

$\det A =$ det a

Advice

For simplicity, we will use the expansion of the first row to find the determinant

$\begin{aligned} \det{A} &= a_1A_1 + b_1B_1+ c_1C_1 \\ &= \var{matrixA[0][0]} \begin{vmatrix} \var{matrixA[1][1]} & \var{matrixA[1][2]} \\ \var{matrixA[2][1]} & \var{matrixA[2][2]} \\ \end{vmatrix} - \var{matrixA[0][1]} \begin{vmatrix} \var{matrixA[1][0]} & \var{matrixA[1][2]} \\ \var{matrixA[2][0]} & \var{matrixA[2][2]} \\ \end{vmatrix} + \var{matrixA[0][2]} \begin{vmatrix} \var{matrixA[1][0]} & \var{matrixA[1][1]} \\ \var{matrixA[2][0]} & \var{matrixA[2][1]} \\ \end{vmatrix} \\ &= \var{matrixA[0][0]} \times \var{cof11} - \left(\var{matrixA[0][1]} \times \var{cof12}\right) + \var{matrixA[0][2]} \times \var{cof13} \\ &= \var{deta} \end{aligned}$

Given arbitrary matrix

$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & j \\ \end{pmatrix}$

It's cofactors are given by

$\begin{aligned} A_{11} &= +\begin{vmatrix} e & f \\ h & j \\ \end{vmatrix} \\ &= +\begin{vmatrix} \var{a22} & \var{a23} \\ \var{a32} & \var{a33} \\ \end{vmatrix} \\ &= \var{cof11} \end{aligned}$

$\begin{aligned} A_{12} &= -\begin{vmatrix} d & f \\ g & j \\ \end{vmatrix} \\ &= -\begin{vmatrix} \var{a21} & \var{a23} \\ \var{a31} & \var{a33} \\ \end{vmatrix} \\ &= \var{cof12} \end{aligned}$

$\begin{aligned} A_{13} &= +\begin{vmatrix} d & e \\ g & h \\ \end{vmatrix} \\ &= +\begin{vmatrix} \var{a21} & \var{a22} \\ \var{a31} & \var{a32} \\ \end{vmatrix} \\ &= \var{cof13} \end{aligned}$

$\begin{aligned} A_{21} &= -\begin{vmatrix} b & c \\ h & j \\ \end{vmatrix} \\ &= -\begin{vmatrix} \var{a12} & \var{a13} \\ \var{a32} & \var{a33} \\ \end{vmatrix} \\ &= \var{cof21} \end{aligned}$

$\begin{aligned} A_{22} &= +\begin{vmatrix} a & c \\ g & j \\ \end{vmatrix} \\ &= +\begin{vmatrix} \var{a11} & \var{a13} \\ \var{a31} & \var{a33} \\ \end{vmatrix} \\ &= \var{cof22} \end{aligned}$

$\begin{aligned} A_{23} &= -\begin{vmatrix} a & b \\ g & h \\ \end{vmatrix} \\ &= -\begin{vmatrix} \var{a11} & \var{a12} \\ \var{a31} & \var{a32} \\ \end{vmatrix} \\ &= \var{cof23} \end{aligned}$

$\begin{aligned} A_{31} &= +\begin{vmatrix} b & c \\ e & f \\ \end{vmatrix} \\ &= +\begin{vmatrix} \var{a12} & \var{a13} \\ \var{a22} & \var{a23} \\ \end{vmatrix} \\ &= \var{cof31} \end{aligned}$

$\begin{aligned} A_{32} &= -\begin{vmatrix} a & c \\ d & f \\ \end{vmatrix} \\ &= -\begin{vmatrix} \var{a11} & \var{a13} \\ \var{a21} & \var{a23} \\ \end{vmatrix} \\ &= \var{cof32} \end{aligned}$

$\begin{aligned} A_{33} &= +\begin{vmatrix} a & b \\ d & e \\ \end{vmatrix} \\ &= +\begin{vmatrix} \var{a11} & \var{a12} \\ \var{a21} & \var{a22} \\ \end{vmatrix} \\ &= \var{cof33} \end{aligned}$

Using our answer from the previous question, we simply write the cofactors in the form

$\begin{pmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \\ \end{pmatrix}$

Giving us our matrix of cofactors

$\begin{pmatrix} \var{cof11} & \var{cof12} & \var{cof13} \\ \var{cof21} & \var{cof22} & \var{cof23} \\ \var{cof31} & \var{cof32} & \var{cof33} \\ \end{pmatrix}$

The transposition process turns rows into columns and columns into rows

Carrying out this process on our matrix of cofactors gives us the adjugate

$\begin{pmatrix} \var{cof11} & \var{cof21} & \var{cof31} \\ \var{cof12} & \var{cof22} & \var{cof32} \\ \var{cof13} & \var{cof23} & \var{cof33} \\ \end{pmatrix}$

We can find the inverse of $A$ using our determinant and adjugate, using the formula

$A^{-1} = \frac{1}{\det A}(adj \; A)$

Therefore, we can calculate $A^{-1}$ by

$\begin{aligned} A^{-1} &= \frac{1}{\var{deta}} \begin{pmatrix} \var{cof11} & \var{cof21} & \var{cof31} \\ \var{cof12} & \var{cof22} & \var{cof32} \\ \var{cof13} & \var{cof23} & \var{cof33} \\ \end{pmatrix} \\ &= \var{inverseA} \end{aligned}$

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This question is used in the following exam:

3x3 Matrices by Fraser Buxton in Tamsin's workspace.

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Inverse of a 3x3 matrix

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Tamsin Smith 7 months, 1 week ago

Tamsin Smith 9 months, 3 weeks ago

Inverse of a $3 \times 3$ Matrix

Error in variable testing condition

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Generated value: `integer`

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Inverse of a 3x3 matrix

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Taxonomy: mathcentre

Taxonomy: Kind of activity

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Contributors

History

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Checkpoint description

Tamsin Smith 7 months, 1 week ago

Tamsin Smith 9 months, 3 weeks ago

Inverse of a 3×33 \times 3 Matrix

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Generated value: integer

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Inverse of a $3 \times 3$ Matrix

Generated value: `integer`