1. ENGG2013 Unit 9 3x3 Determinant
Feb, 2011.

2. Last time 22 determinant
Compute the area of a parallelogram by determinant
A formula for 2x2 matrix inverse
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3. Today 33 determinant and its properties Using determinant, we can
test whether three vectors lie on the same plane
solve 33 linear system
test whether the inverse of a 33 matrix exists
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4. Vector Notation
We will use two different notations for a point in the 3D space
(x,y,z) z z y y x x
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5. 22 determinant Notation for 22 determinant : - bc – ad +
How to calculate: ad – bc
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6. 33 determinant Notation for 33 determinant : Definition: kshum
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7. Rule of Sarrus – – + + – + Pierre Frédéric Sarrus (1798 – 1861)
\begin{array}{|ccc|cc}
a_1 & b_1 & c_1 & {\color{blue} a_1} & {\color{blue} b_1}\\
a_2 & b_2 & c_2 & {\color{blue} a_2} &{\color{blue} b_2}\\
a_3 & b_3 & c_3 & {\color{blue} a_3} &{\color{blue} b_3}\\
\end{array}
Pierre Frédéric Sarrus (1798 – 1861)
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8. Volume of parallelepiped
Geometric meaning
The magnitude of 33 determinant is the volume of a parallelepiped z y x
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9. Co-planar  zero determinant
Determinant = 0  Volume = 0
 the three vectors lie on the same plane z y
A collection of vectors are said to be co-planar if they lie on the same plane. x
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10. Det of Diagonal matrix Volume of a rectangular box c b a kshum
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11. Transpose has the same determinant– – + + + –
Compare with
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12. Volume of parallelepiped
In computing the volume of a parallelepiped, it does not matter whether we write the vector horizontally or vertically in the determinant z
Volume of parallelepiped with vertices (0,0,0), (1,2,0), (2,0,1), (–1, 1, 3) equals to the absolute value of y or x
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13. Question
Do (1,1,1), (2,3,4), (5,6,7) and (8,9,10) lie on the same plane?
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14. Cramer’s rule
If the determinant of a 33 matrix A is non-zero, we can solve the linear system A x = b by Cramer’s rule.
The solution to is
or equivalently A x b
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
x_1\\
x_2\\
x_3 =
b_1\\
b_2\\
b_3
Gabriel Cramer ( )
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15. PROPERTIES OF DETERMINANT
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16. How to show that Cramer’s rule does give the correct answer?
The Cramer’s rule is a theorem, which requires a proof, or verification.
We need some properties of determinant.
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17. Properties of determinant
Taking transpose does not change the value of determinant
We have already verified this property in p.11
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18. Meta-property Because
After taking the transpose of a matrix, columns become rows, and rows become column.
Taking the transpose of a matrix does not change the value of its determinant.
Therefore, any row property of determinant is automatically a column property, and vice versa.
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19. Properties of determinant
If any row or column is zero, then the determinant is 0.
For example
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20. Properties of determinant
If any two columns (or rows) are the identical, then the determinant is zero.
For example, if the second column and the third column are the same, then
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21. Properties of Determinant
If we exchange of the two columns (or two rows), the determinant is multiplied by –1.
For example, if we exchange the column 2 and column 3, we have
The first kind of elementary row operation
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22. Multiply by a constant
If we multiply a row (or a column) by a constant k, the value of determinant increases by a factor of k.
For example, if we multiply the third row by a constant k,
The 2nd kind of elementary row operation
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23. An additive property
If a row (or column) of a determinant is the sum of two rows (or columns), the determinant can be split as the sum of two determinants
For example, if the first column is the sum of two column vectors, then we have
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24. Properties of Determinant
If we add a constant multiple of a row (column) to the other row (column), the determinant does not change.
For example, if we replace the 3rd column by the sum of the 3rd column and k times the 2nd column,
The 3rd kind of elementary row operation
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25. Summary on the effect of the elementary row (or column) operations on determinant
Exchange two rows (or columns)  change the sign of determinant
Multiply a row (or a column) by a constant k  multiply the determinant by k
Add a constant multiple of a row (column) to another row (or column)  no change
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26. Proof of the Cramer’s rule
The solution to is
Verification for x1: Substitute the value of b1, b2 and b3 in the first column of A.
Verification for x2: Substitute the value of b1, b2 and b3 in the second column of A.
Etc.
\begin{vmatrix}
a_{1} & b_1 & c_{1}+kb_1 \\
a_2 & b_2 & c_2 +kb_2\\
a_{3} & b_3 & c_{3}+kb_3
\end{vmatrix} =
a_{1} & b_1 & c_{1} \\
a_2 & b_2 & c_2 \\
a_{3} & b_3 & c_{3}
Cramer’s rule in wikipedia
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27. Because x1, x2, x3 satisfy the system of linear equations, we have
By substitution
Property 6
\begin{vmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{vmatrix}
+ x_2
a_{12}& a_{12} & a_{13}\\
a_{22} & a_{22} & a_{23}\\
a_{32}& a_{32} & a_{33}
\end{vmatrix}+ x_3
a_{13} & a_{12} & a_{13}\\
a_{23} & a_{22} & a_{23}\\
a_{33} & a_{32} & a_{33}
Property 5 =0
By Property 3
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28. MINOR AND COFACTOR
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29. Another way to compute det
Group the six terms as
33 determinant can be computed in terms of 22 determinant
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30. Minor and cofactor
A minor of a matrix is the determinant of some smaller square matrix, obtained by removing one or more of its rows and columns.
Notation: Given a matrix A, the minor obtained by removing the i-th row and j-th column is denoted by Aij. It is also called the minor of the (i,j)-entry aij in A.
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31. Expansion on the first row
\begin{vmatrix}
a_1 & b_1 & c_1\\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3
\end{vmatrix}=a_{1}\begin{vmatrix}
b_2 & c_2\\
b_3 & c_3
\end{vmatrix} - b_1\begin{vmatrix}
a_2 & c_2\\
a_3 & c_3
\end{vmatrix}+c_1
a_2 & b_2\\
a_3 & b_3
\end{vmatrix} \\
Minor of a1
Minor of b1
Minor of c1
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32. Expansion on the second row
\begin{vmatrix}
a_1 & b_1 & c_1\\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3
\end{vmatrix}= -a_2\begin{vmatrix}
b_1 & c_1\\
b_3 & c_3
\end{vmatrix} + b_2\begin{vmatrix}
a_1 & c_1\\
a_3 & c_3
\end{vmatrix} - c_2
a_1 & b_1\\
a_3 & b_3
\end{vmatrix} \\
Minor of a2
Minor of b2
Minor of c2
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33. Expansion on the third row
\begin{vmatrix}
a_1 & b_1 & c_1\\
a_2 & b_2 & c_2\\
a_3 & b_3 & c_3
\end{vmatrix}= a_3\begin{vmatrix}
b_1 & c_1\\
b_2 & c_2
\end{vmatrix} - b_3\begin{vmatrix}
a_1 & c_1\\
a_2 & c_2
\end{vmatrix} + c_3
a_1 & b_1\\
a_2 & b_2
\end{vmatrix} \\
Minor of a3
Minor of b3
Minor of c3
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34. The sign pattern Expansion on the first row
Expansion on the second row
Expansion on the last row
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35. Column expansion
We have similar recursive formula for determinant by column expansion
For example,
Computation on the third column is easy, because there are lots of zeros.
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36. Cofactor
The minor together with the appropriate  sign is called cofactor.
For
The cofactor of Cij is defined as
Expansion on the i-th row (i=1,2,3):
Expansion on the j-th column (j=1,2,3):
The sign
The minor of aij
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37. A formula for matrix inverse
Suppose that det A is nonzero.
(Beware of the subscripts)
Usually called the adjoint of A
Three steps in computing above formula 1. for i,j = 1,2,3, replace each aij by cofactor Cij 2. Take the transpose of the resulting matrix. 3. divide by the determinant of A.
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38. A Quotation Algebra is but written geometry;
geometry is but drawn algebra.
--- Sophie Germain ( )
L'algèbre n'est qu'une géométrie écrite; la géométrie n'est qu'une algèbre figurée
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