 9.9.1: Exer. 114: Without expanding, explain why the statement is true.
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 9.9.7: Exer. 114: Without expanding, explain why the statement is true.
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 9.9.9: Exer. 114: Without expanding, explain why the statement is true.
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 9.9.13: Exer. 114: Without expanding, explain why the statement is true.
 9.9.14: Exer. 114: Without expanding, explain why the statement is true.
 9.9.15: Exer. 1524: Find the determinant of the matrix after introducing ze...
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 9.9.17: Exer. 1524: Find the determinant of the matrix after introducing ze...
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 9.9.19: Exer. 1524: Find the determinant of the matrix after introducing ze...
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 9.9.24: Exer. 1524: Find the determinant of the matrix after introducing ze...
 9.9.25: Show that (Hint: See Example 3.)
 9.9.26: Show that 1aa31bb31c a bb cc aa b c.1
 9.9.27: If show that A a1100a12a2200a13a23a330a14a24a34a44,.
 9.9.28: If , show that A ac0bd000g00f
 9.9.29: If and are arbitrary square matrices of order 2, show that .
 9.9.30: If is a square matrix of order n and k is any real number, show tha...
 9.9.31: Use properties of determinants to show that the following is an equ...
 9.9.32: Use properties of determinants to show that the following is an equ...
 9.9.33: Exer. 3342: Use Cramers rule, whenever applicable, to solve the sys...
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 9.9.42: Exer. 3942: Express the determinant in the form for real numbers a,...
Solutions for Chapter 9.9: Properties of Determinants
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 9.9: Properties of Determinants
Get Full SolutionsChapter 9.9: Properties of Determinants includes 42 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Since 42 problems in chapter 9.9: Properties of Determinants have been answered, more than 37607 students have viewed full stepbystep solutions from this chapter.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.