Determinant of a 3x3 matrix

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Frank Doheny

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Frank Doheny

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7 years, 3 months ago

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Frank Doheny 7 years, 3 months ago

Gave some feedback: Ready to use

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Frank Doheny 7 years, 3 months ago

Created this as a copy of Inverse of a 3x3 matrix.

Name	Status	Author	Last Modified
Inverse of a 3x3 matrix	Ready to use	Frank Doheny	14/03/2018 21:05
Mark's copy of Inverse of a 3x3 matrix	draft	Mark Hodds	07/09/2017 14:58
Mark's copy of Mark's copy of Inverse of a 3x3 matrix	draft	Mark Hodds	07/09/2017 15:00
Determinant of a 3x3 matrix	Ready to use	Frank Doheny	14/03/2018 21:10
Determinant of a 3x3 matrix	draft	Xiaodan Leng	11/07/2019 00:31
3x3 Determinant Problems	draft	Tamsin Smith	17/09/2024 13:14

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Question statement

Calculate the two values for $x$ that satisfy the equation $|A|=\var{k}$

where the matrix $A$ is given by:

$\mathbf{A}=\begin{pmatrix} x&\var{a}&\var{b}\\ \var{c}&x&\var{d}\\\var{e1}&\var{f}&\var{g} \end{pmatrix}$

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			Name	Type	Generated Value

a	integer	20
b	integer	7
c	integer	12
d	integer	5
e1	integer	6
f	integer	1
g	integer	1
k	integer	234

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Advice

Given the matrix:

$\mathbf{A}=\begin{pmatrix} {a11}&{a12}&{a13}\\ {a21}&{a22}&{a23}\\{a31}&{a32}&{a33} \end{pmatrix}$

The determinant of a 3x3 matrix is determined by the formula:

$|\mathbf{A}|={a11}*\begin{vmatrix}{a22}&{a23}\\{a32}&{a33}\end{vmatrix}-{a12}*\begin{vmatrix}{a21}&{a23}\\{a31}&{a33}\end{vmatrix}+{a13}*\begin{vmatrix}{a21}&{a22}\\{a31}&{a32}\end{vmatrix}$

So in this example:

$\mathbf{A}=\begin{pmatrix}x&\var{a}&\var{b}\\ \var{c}&x&\var{d}\\\var{e1}&\var{f}&\var{g} \end{pmatrix}$

$|\mathbf{A}|=x*\begin{vmatrix}x&\var{d}\\\var{f}&\var{g}\end{vmatrix}-\var{a}*\begin{vmatrix}\var{c}&\var{d}\\\var{e1}&\var{g}\end{vmatrix}+\var{b}*\begin{vmatrix}\var{c}&x\\\var{e1}&\var{f}\end{vmatrix}$

$|\mathbf{A}|=x*(x*\var{g}-\var{d}*\var{f})-\var{a}*(\var{c}*\var{g}-\var{e1}*\var{d})+\var{b}*(\var{c}*\var{f}-\var{e1}x)$

$|\mathbf{A}|=x(x-\simplify{{d}{f}})-\var{a}(\simplify{{c}*{g}}-\simplify{{e1}*{d}})+\var{b}(\simplify{{c}*{f}}-\var{e1}x)$

$|\mathbf{A}|=x^2-\simplify{{d}*{f}}x-\simplify{{a}*{c}*{g}}+\simplify{{a}*{e1}*{d}}+\simplify{{b}*{c}*{f}}-\simplify{{b}*{e1}}x$

$|\mathbf{A}|=x^2-\simplify{{d}*{f}+{b}*{e1}}x+\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}}$

If $|\mathbf{A}|=\var{k}$

$x^2-\simplify{{d}*{f}+{b}*{e1}}x+\simplify{{a}*{e1}*{d}+{b}*{c}*{f}-{a}*{c}*{g}}=\var{k}$

$x^2-\simplify{{d}*{f}+{b}*{e1}}x+\simplify{{b}*{e1}*{f}*{d}}=0$

This can be solved by formula or by finding factors

$(x-\simplify{{f}*{d}})(x-\simplify{{b}*{e1}})=0$

$x-\simplify{{f}*{d}}=0\,\,\,\,\,or\,\,\,\,\,x-\simplify{{b}*{e1}}=0$

$x=\simplify{{f}*{d}}\,\,\,\,\,or\,\,\,\,\,x=\simplify{{b}*{e1}}$

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Frank Doheny 7 years, 3 months ago

Frank Doheny 7 years, 3 months ago

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Determinant of a 3x3 matrix

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Frank Doheny 7 years, 3 months ago

Frank Doheny 7 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`