 3.2.1: Compute the following determin,Ults via reduction to trio angular f...
 3.2.2: Compute the following determinants via reduction to triangular form...
 3.2.3: If b l b, h) =3.find
 3.2.4: f b l c, :: :: I =  2. find
 3.2.5: If ~: b, h) = 4. find
 3.2.6: Verify that del(A S ) = det(A)det(B) for the following:
 3.2.7: Evaluate: (' J 4 2 0 0 2 3 I 0 3 I 0 2 303 o o o 3 2 2 o 0 4 0  I...
 3.2.8: Is det(AB) = det(BA)? Justify your answer
 3.2.9: If del(AB) = O. is del(A) = 0 or det(B) = O'! Gille reasons for you...
 3.2.10: Show that if k is a scalar and A is II X II. then det(kA) = k" de\(A)
 3.2.11: Show 1hal if A is" x" Wilh" odd and skew symmetric. then del (A) = O.
 3.2.12: Show that if A is a matrix such that in each row and in each column...
 3.2.13: Show that det(AB ') = .
 3.2.14: Show that if AB = I, . then det(A) 1= 0 and det(B) 1= O.
 3.2.15: (a ) Show that if A = A '. then det(A) =l. (b) If AT = A  I. whal...
 3.2.16: Show that if A ,md B are square m
 3.2.17: If A is a nonsingular matrix such that A2 = A. what is det(A)?
 3.2.18: Prove Corollary 3.2.
 3.2.19: Show that if A . B. and C
 3.2.20: Show that if A and B are both /I x /I. then (a) det(A T BT) = det(A...
 3.2.21: Verify the result in Exercise 16 for A [~~] and B~[ _~ ;J
 3.2.22: Use the properties of Section 3.2 10 prove that b ,,' b~ I = (b  u...
 3.2.23: If det(A) = 2. find det(As)
 3.2.24: Use Theorem 3.8 10 detennine which of the following matrices are no...
 3.2.25: Use 111eorem 3.8 10 detennine which of the following matrices are n...
 3.2.26: Use Theorem 3.8 to determine all values of I so that the following ...
 3.2.27: Use Corollary 3. 1 to find out whe ther the following homogeneous s...
 3.2.28: Repeat Exercise 27 for the following homogeneom system: [~ 2 I o o 2 I
 3.2.29: Let A [aii ] be an upper triangular matrix. Prove that A is nonsing...
 3.2.30: Let A be a 3 x 3 matrix with det(A) = 3. (a) What is the reduced ro...
 3.2.31: Let A be a 4 x 4 malrix with det(A) = O. (a) Describe the reduced r...
 3.2.32: 1.e.1 A2 = A . det(A) = I.
 3.2.33: Prove Corollary 3.3
 3.2.34: Let AS = AC. Prove Ihat if det(A) 1= O. then B = C.
 3.2.35: Determine whether the software you are using has a command for comp...
 3.2.36: Assuming m;lt your software has a command to con pUle the determina...
 3.2.37: TIlcorem 3.8 assumes thai all calculations for de\(A) are done hy e...
Solutions for Chapter 3.2: Properties of Determinants
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 3.2: Properties of Determinants
Get Full SolutionsChapter 3.2: Properties of Determinants includes 37 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 37 problems in chapter 3.2: Properties of Determinants have been answered, more than 9145 students have viewed full stepbystep solutions from this chapter.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.