 2.1: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.2: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.3: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.4: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.5: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.6: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.7: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.8: In Exercises 18, evaluate the determinant of the given matrix by (a...
 2.9: Evaluate the determinants in Exercises 36 by using the arrow techni...
 2.10: (a) Construct a 4 4 matrix whose determinant is easy to compute usi...
 2.11: Use the determinant to decide whether the matrices in Exercises 14 ...
 2.12: Use the determinant to decide whether the matrices in Exercises 58 ...
 2.13: In Exercises 1315, find the given determinant by any method.5 b 3 b...
 2.14: In Exercises 1315, find the given determinant by any method.3 4 a a...
 2.15: In Exercises 1315, find the given determinant by any method.0000 3 ...
 2.16: Solve for x. x 1 3 1 x = 1 0 3 2 x 6 1 3 x 5
 2.17: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.18: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.19: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.20: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.21: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.22: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.23: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.24: In Exercises 1724, use the adjoint method (Theorem 2.3.6) to find t...
 2.25: Use Cramers rule to solve for x and y in terms of x and y. x = 3 5 ...
 2.26: Use Cramers rule to solve for x and y in terms of x and y. x = x co...
 2.27: By examining the determinant of the coefficient matrix, show that t...
 2.28: Let A be a 3 3 matrix, each of whose entries is 1 or 0. What is the...
 2.29: (a) For the triangle in the accompanying figure, use trigonometry t...
 2.30: Use determinants to show that for all real values of , the only sol...
 2.31: Prove: If A is invertible, then adj(A) is invertible and [adj(A)] 1...
 2.32: Prove: If A is an n n matrix, then det[adj(A)]=[det(A)] n1
 2.33: Prove: If the entries in each row of an n n matrix A add up to zero...
 2.34: (a) In the accompanying figure, the area of the triangle ABC can be...
 2.35: Without directly evaluating the determinant, show that sin cos sin(...
Solutions for Chapter 2: Determinants
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 2: Determinants
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Chapter 2: Determinants includes 35 full stepbystep solutions. Since 35 problems in chapter 2: Determinants have been answered, more than 17042 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. This expansive textbook survival guide covers the following chapters and their solutions.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

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.

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.

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.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.