 3.2.10E: Find the determinants in Exercises by row reduction to echelon form.
 3.2.11E: Combine the methods of row reduction and cofactor expansion to comp...
 3.2.13E: Combine the methods of row reduction and cofactor expansion to comp...
 3.2.1E: Each equation in Exercises 1–4 illustrates a property of determinan...
 3.2.2E: Each equation in Exercises illustrates a property of determinants....
 3.2.3E: Each equation in Exercises illustrates a property of determinants....
 3.2.4E: Each equation in Exercises illustrates a property of determinants....
 3.2.5E: Find the determinants in Exercises by row reduction to echelon form.
 3.2.6E: Find the determinants in Exercises by row reduction to echelon form.
 3.2.7E: Find the determinants in Exercises 5–10 by row reduction to echelon...
 3.2.8E: Find the determinants in Exercises by row reduction to echelon form.
 3.2.9E: Find the determinants in Exercises by row reduction to echelon form.
 3.2.14E: Combine the methods of row reduction and cofactor expansion to comp...
 3.2.15E: Find the determinants in Exercises, where
 3.2.16E: Find the determinants in Exercises, where
 3.2.17E: Find the determinants in Exercises, where
 3.2.18E: Find the determinants in Exercises, where
 3.2.19E: Find the determinants in Exercises 15–20, where
 3.2.20E: Find the determinants in Exercises, where
 3.2.21E: In Exercises, use determinants to find out if the matrix is inverti...
 3.2.22E: In Exercises, use determinants to find out if the matrix is inverti...
 3.2.24E: In Exercises, use determinants to decide if the set of vectors is l...
 3.2.25E: In Exercises 24–26, use determinants to decide if the set of vector...
 3.2.26E: In Exercises, use determinants to decide if the set of vectors is l...
 3.2.27E: In Exercises 27 and 28, A and B are n × n matrices. Mark each state...
 3.2.28E: A and B are n × n matrices. Mark each statement True or False. Just...
 3.2.29E: Compute det B4 where B =
 3.2.30E: Use Theorem 3 (but not Theorem 4) to show that if two rows of a squ...
 3.2.31E: In Exercises 31–36, mention an appropriate theorem in your explanat...
 3.2.32E: Suppose that 4 is a square matrix such that detA3 = 0. Explain why ...
 3.2.33E: In Exercises 31–36, mention an appropriate theorem in your explanat...
 3.2.34E: In Exercises 31–36, mention an appropriate theorem in your explanat...
 3.2.35E: In Exercises 31–36, mention an appropriate theorem in your explanat...
 3.2.36E: In Exercises 31–36, mention an appropriate theorem in your explanat...
 3.2.37E: Verify that dt AB = (det A)(det B) for the matrices in Exercises 37...
 3.2.38E: Verify that det AB = (det A)(det B) for the matrices in Exercises.A...
 3.2.39E: Let A and B be 3 × 3 matrices, with det A = —3 anddet B = 4. Use pr...
 3.2.40E: Let A and B be 4 × 4 matrices, with det A = —3 anddet B = — 1. Comp...
 3.2.41E: Verify that det A = det B + det C, where
 3.2.42E: Show that
 3.2.43E: Verify that det A = det B + det C, where
 3.2.44E: Rightmultiplication by an elementary matrix E affects the columns ...
 3.2.45E: [M] Compute det ATA and detAAT for several random 4 × 5 matrices an...
Solutions for Chapter 3.2: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 3.2
Get Full SolutionsChapter 3.2 includes 43 full stepbystep solutions. Since 43 problems in chapter 3.2 have been answered, more than 25793 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.