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• Invertible matrix (a.k.a. non-singular matrix)– Represents reversible linear transformation
• Gauss-Jordan elimination– Algorithm for compute matrix inverse
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Carl Friedrich Gauss (1777 – 1853)
• Borned: Braunschweig, Germany.• Died: Göttingen, Germany.• Great mathematician in the 18th century.• Legacy
• Gaussian distribution• Gaussian elimination• Gauss-Jordan elimination• Gaussian curvature in differential geometry• Proof of the quadratic reciprocity in number
theory• Construction of 17-gon by straightedge and
compass• And much more
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Wilhelm Jordan (1842 –1899)
• Borned: Ellwangen, Germany.• Died: Hanover, Germany.• Geodesist.• Remembered for:
– Surveying in Germany and Africa
– His book “Textbook of Geodesy” (Handbuch der Vermessungskunde) popularizes the Gauss-Jordan algorithm
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22 Determinant
• Area of parallelogram
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(a,b)
(c,d)
Row 1 is the first vectorRow 2 is the second vector
d
b
c
a
22 Determinant
• Area of parallelogram
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(3,1)
(2,4)
Row 1 is the first vectorRow 2 is the second vector
Area = 10
Outline
• Computing 22 matrix inverse via determinant
• Properties of determinant• Explain why the absolute value of determinant
is the area of parallelogram?
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Determinant of 22 matrix
• Notation: Given a 22 matrix we use the notation
to stand for the determinant ps – qr.
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or
A formula for the Inverse of a 2x2 matrix
• Given 22 matrix
• Want to find the inverse of A
• A formula for A-1: If det A is nonzero, we have
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How to compute the inverseof a 2x2 matrix
1. Exchange the two diagonal entries a, d.2. Take the negative of the two off-diagonal
entries b, c.3. Divide by the determinant.kshum ENGG2013 12
Application in solving equations
• Solve
• If we know the inverse of the 2x2 matrix, we can solve the linear system easily.
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Properties of determinant
1. A matrix and its transpose has the same determinant
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“The transpose of a matrix”means reflecting the matrixalong the diagonal.
Row 1 and row 2 becomecolumn 1 and column 2, andvice versa.
We write AT for the transpose of matrix A.
Proof is obvious, because
Meta-property
Any row property of determinant is a column
property, and vice versa
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By property 1, we have:
Properties of determinant
2. If any row or column is zero, then the determinant is 0.
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Zero area
Properties of determinant
3. If the two columns (or two rows) are constant multiple of each other, the determinant is zero.
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Zero area
Properties of Determinant
4. If we exchange of the two columns (or two rows), the determinant is multiplied by –1.
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(2,1)
(1,3)
The first kindof elementaryrow operation
Properties of Determinant
5. If we multiply a row (or a column) by a constant c, the value of determinant also increase by a factor of c.
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The 2nd kindof elementaryrow operation
(0,1) (1,1)
(4,4)
Properties of Determinant
6. If we add a constant multiple of a row (column) to the other row (column), the determinant does not change.
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The 3rd kindof elementaryrow operation
(1,0)
(0,1) (3,1)
Properties of Determinant
7. In the linear transformation represented by a 22 matrix, the magnitude of determinant measures the area expanding factor.
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Multiplied by
Square withunit area
= ad – bcThe ratio of area
(0,1) (1,1)
(1,0)
Properties of Determinant8. For any 2x2 matrices A and B, we have the
following multiplicative property
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Multiplied by A
Square withunit area
Parallelogramarea = det(A)
Expand by a factor of det(A)
Parallelogramarea = det(AB)= det(A) det(B)
Multiplied by B
Expand by a factor of det(B)
Expand by a factor of det(AB) = det(A) det(B)
Determinant as area
• Using the properties in previous pages, we are now ready to show that the absolute value of det(M) is equal to the area of parallelogram whose sides are the two rows of M.
• We divide the argument into two steps– The two rows are perpendicular (special case).– The two rows are not perpendicular (general
case).
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Determinant as area (I)• Suppose that the two rows are perpendicular i.e., ac+bd = 0 (dot product of [a b] and [c d] are zero)• Let
• Want to show that
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(a,b)(c,d)
The trick is to show instead.
Determinant as area (I)
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By Property 1
Just write down M and MT
By Property 8
By the definition ofmatrix multiplication
Because the dot productac+bd is zero by assumption
By the definition of determinant
By Pythagoras theorem, are the height and width of the rectangle.
Determinant as area (II)• Suppose that the two rows of M are not perpendicular.
• Idea: “Slide” the parallelogramto a rectangle, whilekeeping the areaunchanged.
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(a,b)
(c,d)
Determinant as Area (II)
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Decompose [c d] into two components, one is perpendicular to [a b],and the other along the same direction as [a b].
(a,b)
(c,d)
Find the constant k (hoemwork exercise)
Perpendicular to [a b](The height of parallelogram)
In the same direction as [a b]
Determinant as area (II)
• Choose a constant k such that
• Let• [c’ d’] is perpendicular to [a b] by our choice of
k.• By definition the area of parallelogram is
equal to• But
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By property 6 By the first part of our proof in p.27