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4. Determinants4. Determinants
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Determinants of 2×2 and 3×3 Matrices
2×2 determinant
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Determinants of 2×2 and 3×3 Matrices
3×3 determinant
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Elementary Products
(4)
Elementary product: a product containing one entry from each row and one entry from each column.
In formula (4) each elementary product is of the form
where the blank contain some permutation of the column indices {1, 2, 3}.
The sign can be determined by counting the minimum number of interchanges in the permutation of the column indices required to put those indices into their natural order: the sign is + if the number is even and – if it is odd.
Signed elementary product
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Elementary Products
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
General Determinants
n×n determinant or n-th order determinant
The signed elementary products are to be summed over all possible permutations {j1, j2, …, jn} of the column indices.
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Determinants of Matrices with Rows or Columns That Have All Zeros
Determinants of Triangular Matrices
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Minors and Cofactors
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Minors and Cofactors
Example 3Example 3
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Cofactor Expansions
These are called cofactor expansions of A. Note that in each cofactor expansion, the entries and cofactors all come from the same row or the same column.
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Cofactor Expansions
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4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion
Cofactor Expansions
Example 5Example 5Use a cofactor expansion to find the determinant of
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4.2. Properties of Determinants4.2. Properties of Determinants
Determinant of AT
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4.2. Properties of Determinants4.2. Properties of Determinants
Effect of Elementary Row Operations on A Determinant
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4.2. Properties of Determinants4.2. Properties of Determinants
Effect of Elementary Row Operations on A Determinant
Suppose that the ith row of A is multiplied by the scalar k to produce the matrix B,
Since the ith row is deleted when the cofactors along that row are computed, the cofactors in this formula are unchanged when the ith row is multiplied by k.
(a)
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4.2. Properties of Determinants4.2. Properties of Determinants
Effect of Elementary Row Operations on A Determinant
Example 1Example 1
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4.2. Properties of Determinants4.2. Properties of Determinants
Effect of Elementary Row Operations on A Determinant
Example 2Example 2
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4.2. Properties of Determinants4.2. Properties of Determinants
Effect of Elementary Row Operations on A Determinant
Example 3Example 3What is the relationship between det(A) and det(-A)?
Thus, det(-A)=det(A) if n is even, and det(-A)=-det(A) if n is odd.
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4.2. Properties of Determinants4.2. Properties of Determinants
Simplifying Cofactor Expansions
(1) A cofactor expansion can be minimized by expanding along a row or column with the maximum number of zeros.
(2) Adding multiples of one row (or column) to another does not change the determinant of the matrix.
Example 4Example 4
Use a cofactor expansion to find the determinant of
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4.2. Properties of Determinants4.2. Properties of Determinants
A Determinant Test for Invertibility
If R is the reduced row echelon form of square matrix A, determinant of the matrices are both zero or both nonzero.
Assume that A is invertible, in which case the reduced row echelon form of A is I (by Theorem 3.3.3). Since det(I)≠0, it follows that det(A)≠0.
Conversely, assume that det(A)≠0. Since det(R)≠0, R=I. Thus, A is invertible (by Theorem 3.3.3)
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4.2. Properties of Determinants4.2. Properties of Determinants
Determinants of A Product of Matrices
Determinants of The Inverse of A Matrix
Since AA-1=I, det(AA-1)=det(I)=1=det(A)det(A-1). Thus, det(A-1)=1/det(A).
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4.2. Properties of Determinants4.2. Properties of Determinants
Determinants of A+B
det(A+B)≠det(A)+det(B)
Example 8Example 8
det(A)=1, det(B)=5, det(A+B)=23.
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4.2. Properties of Determinants4.2. Properties of Determinants
Determinant Evaluation by LU-Decomposition
If A=LU, then det(A)=det(L)det(U), which is easy to compute since L and U are triangular.
Thus, nearly all of the computational work in evaluating det(A) is expended in obtaining the LU-decomposition.
From Table 3.7.1.The number of flops required to obtain the LU-decomposition of an n×n matrix is on the order of 2/3n3 for large values of n.This is an enormous improvement over the determinant definition, which involves the computation of n! signed elementary products.
For example, today’s typical PC can evaluate 30×30 determinant in less than one-thousandth of a second by LU-decomposition compared to the roughly 1010 years that would be required for it to evaluate 30! Signed elementary products.
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4.2. Properties of Determinants4.2. Properties of Determinants
A Unifying Theorem
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4.3. Cramer4.3. Cramer’’s Rules Rule
Adjoint of A Matrix
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4.3. Cramer4.3. Cramer’’s Rules Rule
Adjoint of A Matrix
If we multiply the entries in the first row by the corresponding cofactors from the third row, then the sum is
Let’s consider a matrix A’ that results when the third row of A is replaced by a duplicate of the first row.
We know that det(A’)=0 because of the duplicate rows.
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4.3. Cramer4.3. Cramer’’s Rules Rule
Adjoint of A Matrix
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4.3. Cramer4.3. Cramer’’s Rules Rule
Adjoint of A Matrix
Example 1Example 1Find the matrix of cofactors from A.
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4.3. Cramer4.3. Cramer’’s Rules Rule
A Fomula for The Inverse of A Matrix
Suppose that A is invertible.
The entry in the ith row and jth column of this product is
(3)
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4.3. Cramer4.3. Cramer’’s Rules Rule
A Fomula for The Inverse of A Matrix
In the case where i=j, the entries and cofactors come from the same row of A, so (3) is the cofactor expansion of det(A) along that row.
In the case where i≠j, the entries and cofactors come from different rows, so the sum is zero by Theorem 4.3.1.
Since A is invertible, it follows that det(A)≠0, so this equation can be rewritten as
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4.3. Cramer4.3. Cramer’’s Rules Rule
A Fomula for The Inverse of A Matrix
Example 2Example 2Find the inverse of the matrix A in Example 1.
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4.3. Cramer4.3. Cramer’’s Rules Rule
How The Inverse Formula Is Used
Computer program usually use LU-decomposition (as discussed in Section 3.7) and not Formula (2) to invert matrices.
Thus, the value of Formula (2) is not for numerical computations, but rather as a tool in theoretical analysis.
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cramer’s Rule
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cramer’s Rule
In the case where det(A)≠0, we can use Formula (2) to rewrite the unique solution of Ax=b as
Therefore, the entry in the jth row of x is
The cofactors in this expression come from the jth column of A and hence remain unchanged if we replace the jth column of A by b (the jth column is crossed out when the cofactors are computed). Since this substitution produces the matrix Aj, the numerator in (5) can be interpreted as the cofactor expansion along the jth column of Aj. Thus,
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cramer’s Rule
Example 4Example 4Use Cramer’s rule to solve the system
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cramer’s Rule
Example 5Example 5Use Cramer’s rule to solve the system
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4.3. Cramer4.3. Cramer’’s Rules Rule
Geometric Interpretation of Determinants
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4.3. Cramer4.3. Cramer’’s Rules Rule
Geometric Interpretation of Determinants
(a)
Suppose that the matrix A is partitioned into columns as
and let us assume that the parallelogram with adjacent sides u and v is not degenerate.
We can see that this area can be expressed as
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4.3. Cramer4.3. Cramer’’s Rules Rule
Geometric Interpretation of Determinants
The square of the area can be expressed as
(area)2=
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4.3. Cramer4.3. Cramer’’s Rules Rule
Geometric Interpretation of Determinants
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4.3. Cramer4.3. Cramer’’s Rules Rule
Geometric Interpretation of Determinants
Example 8Example 8Find the area of the triangle with vertices A(-5, 4), B(3, 2), and C(-2, -3).
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
Standard unit vector i=(1,0,0), j=(0,1,0), k=(0,0,1)
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
Example 9Example 9Let u=(1,2,-2) and v=(3,0,1). Find
(a) u×v (b) v×u (c) u×u
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
In general, if u and v are nonzero and nonparallel vectors, then the direction of u×v in relation to u and v can be determined by the right-hand rule.
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
The associative law does not hold for cross products; for example,
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
(a) Since 0≤ θ ≤ π, it follows that sin θ ≥0 and hence that
Thus,
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4.3. Cramer4.3. Cramer’’s Rules Rule
Cross Products
(b)
Example 10Example 10
Find the area of the triangle in R3 that has vertices P1(2,2,0), P2(-1,0,2), and P3(0,4,3).
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Fixed Points
Recall that a fixed point of an n×n matrix A is a vector x in Rn such that Ax=x (see the discussion preceding Example 6 of Section 3.6)
Every square matrix A has at least one fixed point, namely x=0. We call this the trivial fixed point of A.
Ax=x Ax=Ix (A-I)x=0
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
One might also consider more general equations of the form Ax= λx in which λ is a scalar.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
The most direct way of finding the eigenvalues of an n×n matrix A is to rewrite the equation Ax= λx as Ax= λIx, or equivalently, as
and then try to determine those values of λ, if any, for which this system has nontrivial solutions. Since (4) has nontrivial solutions if and only if the coefficient matrix λI-A is singular, we see that the eigenvalues of A are the solutions of the equation
(4)
This is called the characteristic equation of A. Also, if λ is an eigenvalue of A, then (4) has a nonzero solution space, which we call the eigenspace of A corresponding to λ.
It is the nonzero vectors in the eigenspace of A corresponding to λ that are the eigenvectors of A corresponding to λ.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
Example 2Example 2(a) Find the eigenvalues and corresponding eigenvectors of the matrix
(b) Graph the eigenspaces of A in an xy-coordiante system.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalues of Triangular Matrices
If A is an n×n triangular matrix with diagonal entries a11, a22, …, ann, then λI-A is a triangular matrix with diagonal entries λ-a11, λ-a22, …, λ-ann. Thus the characteristic polynomial of A is
which implies that the eigenvalues of A are
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalues of Powers of A Matrix
If λ is an eigenvalue of A and x is a corresponding eigenvector, then
which shows that λ2 is an eigenvalue of A2 and x is a corresponding eigenvector.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
A Unifying Theorem
λ=0 is an eigenvalue of A if and only if there is a nonzero vector x such that Ax=0.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Algebraic Multiplicity
REMARK
This theorem implies that an n×n matrix has n eigenvalues if we agree to count repetitions and allow complex eigenvalues, but the number of distinct eigenvalues may be less than n.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalue Analysis of 2×2 Matrices
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalue Analysis of 2×2 Symmetric Matrices
Later in the text we will show that all symmetric matrices with real entries have real eigenvalues.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalue Analysis of 2×2 Symmetric Matrices
If the 2×2 symmetric matrix is
then
so Theorem 4.4.9 implies that A has real eigenvalues.
It also follows from that theorem that A has one repeated eigenvalue if and only if
Since this holds if and only if a=d and b=0.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalue Analysis of 2×2 Symmetric Matrices
Example 6Example 6Graph the eigenspaces of the symmetric matrix
in an xy-coordinate system.
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Expressions for Determinant and Trace in Terms of Eigenvalues
(a)
Write the characteristic polynomial in factored form:
Setting λ=0 yields
But det(-A)=(-1)ndet(A), so it follows that
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Expressions for Determinant and Trace in Terms of Eigenvalues
(b)
Assume that A=[aij], so we can write p(λ) as
Any elementary product that contains an entry that is off the main diagonal of (30) as a factor will contain at most n-2 factors that involve λ. Thus the coefficient of λn-1 in p(λ) is the same as the coefficient of λn-1 in the product
(30)
Expanding this product we see that it has the form
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Expressions for Determinant and Trace in Terms of Eigenvalues
and expanding the expression for p(λ) we see that it has the form
=
Thus, we must have
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Expressions for Determinant and Trace in Terms of Eigenvalues
Example 7Example 7Find the determinant and trace of a 3×3 matrix whose characteristic polynomial is
This polynomial can be factored as
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4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors
Eigenvalues by Numerical Methods
Eigenvalues are rarely obtained by solving the characteristic equation in real-world applications primarily for two reasons:
1. In order to construct the characteristic equation of an n×n matrix A, it is necessary to expand the determinant det(λI-A). The computations are prohibitive for matrices of the size that occur in typical applications.
2. There is no algebraic formula of finite algorithm that can be used to obtain the exact solutions of the characteristic equation of a general n×n matrix when n≥5.
Given these impediments, various algorithms have been developed for producing numerical approximations to the eigenvalues and eigenvectors.