Find determinant of a 3x3 matrix by row reduction

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

You are given the matrix $\mathrm{M} = \var{m3}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

m1	matrix	Matrix of size 3×3
c11	integer	300
c12	integer	100
m2	matrix	Matrix of size 3×3
c20	integer	1
m3	matrix	Matrix of size 3×3
determinant	number	-12
d1	matrix	Matrix of size 3×3
upper_triangular	matrix	Matrix of size 3×3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
matrix(repeat(repeat(random(1..4),3),3))

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `matrix`

matrix([1,4,4],[2,2,2],[1,1,3])

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d1
determinant
m2

Advice
"Part a)" - Correct answer
"Part c)" - Minimum accepted value
"Part c)" - Maximum accepted value

Matrix entry

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

Write another matrix of single-digit values (each element of the matrix should be an integer in the range $[-9,9]$ ) whose determinant is the same as $\mathrm{M}$ .

Marks

Correct answer

xxxxxxxxxx
 
m1

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Number of columns

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Margin of error allowed in each cell

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Precision restriction

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Display numbers in the correct answer as fractions?

Pre-filled cells

​x
 
​

Advice

a)

Note that subtracting one row from another does not affect the determinant. In fact, adding or subtracting any multiple of one row from another does not affect the determinant. That is, the row operation $r_i \to r_i + ar_j$ , $i \neq j$ , does not affect the determinant.

Subtract $\simplify{{c20}r_2}$ from $r_1$ to get $\mathrm{M}_1 = \var{m2}$ .

Subtract $\simplify{{c11}r_1}$ from $r_2$ and $\simplify{{c12}r_1}$ from $r_3$ of $\mathrm{M}_1$ to get $\mathrm{M}_2 = \var{m1}$ .

b)

Considering each column in turn, we use row operations to ensure that there are zeros in each entry below the main diagonal, $\mathrm{M}_{ij}$ , $i \gt j$ .

Subtract $\simplify[fractionnumbers,unitfactor]{{m1[1][0]/m1[0][0]}r_1}$ from $r_2$ and $\simplify[fractionnumbers,unitfactor]{{m1[2][0]/m1[0][0]}r_1}$ from $r_3$ of $\mathrm{M}_2$ to obtain $\mathrm{M}_3 = \simplify[fractionnumbers]{{d1}}$ . The first entry of each row, apart from the first row, is zero.

Subtract $\simplify[fractionnumbers,unitfactor]{{m1[2][1]/m1[1][1]}r_2}$ from $r_3$ of $\mathrm{M}_3$ to obtain $\mathrm{M}_4 = \simplify[fractionnumbers]{{upper_triangular}}$ . The second entry of each row, apart from the second row, is zero.

c)

Using the upper triangular matrix found in part b), the determinant of $\mathrm{M}$ is $\simplify[!basic,fractionnumbers]{{upper_triangular[0][0]}*{upper_triangular[1][1]}*{upper_triangular[2][2]}} = \var{determinant}$ .

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Find determinant of a 3x3 matrix by row reduction

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

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Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `matrix`

Parts

Matrix entry

Advanced settings

a)

b)

c)

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Error

Find determinant of a 3x3 matrix by row reduction

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: matrix

Parts

Matrix entry

Advanced settings

a)

b)

c)

Generated value: `matrix`