 1.5.1: (a) Show th:lt if A is any III )( n matrix. then I", A = A and A I"...
 1.5.2: Prove that the sum. product, and scalar multiple of diag. onal. sca...
 1.5.3: Prove: If A and 8 are I! x II diagonal matrices. then AB = BA.
 1.5.4: Let  3 2 o Verify that A + 8 and A B are upper triangu lar.
 1.5.5: Describe:lll matrices that are both upper :lnd lower trian gular
 1.5.6: LetA=[~ ~~]and =[~ ~lcomputeeach of the following
 1.5.7: Lo< A ~ and S 0 Compute e:lch of the following: (a) A3 (b ) S! ) (A...
 1.5.8: Let p and q be nonnegative integers and let A be:l square matrix. S...
 1.5.9: If AS = BA and p is a nonnegative integer. show that {AB)P = A PB "
 1.5.10: If p is a nonneg:ltive integer and e is a scalar. show that (eA)!' ...
 1.5.11: For:l square matri x A and:l nonnegative integer p. show [hat (AT)"...
 1.5.12: For a nonsingul:lr matrix A and a nonneg:ltive integer p . show tha...
 1.5.13: For a nonsingular matrix A :lnd nonzero scabr k. show that (kA) 1 ...
 1.5.14: (a) Show that every sC:llar matrix is symmetric. (b) Is every scala...
 1.5.15: Find a 2 x 2 matrix 8 fAS= BA where A = [' . 2 ces S are there?
 1.5.16: Find a 2 x 2 matrix B fAB=BA. Where A= [~ ces B are there?
 1.5.17: Prove or disprove: For any 11 XII matrix A. AT A = AAT.
 1.5.18: (a) Show tlwt A is symmetric if and only if (Ii) = {I i i furalli.j...
 1.5.19: Show that if A is a symmetric matrix. then AT is symmetric
 1.5.20: Describe all skew syr.lmetric scalar m:ltrices
 1.5.21: Show that if A is any III x n matrix. then AA T and AT A alC SY Ill...
 1.5.22: Show that if A is any I! x I! matrix . then (a) A + A T is symmetri...
 1.5.23: Show that if A is a symmetric m:ltrix, then A I, k 2.3 ..... is sym...
 1.5.24: Let A and S be symmetric m:ltrices. (a) Show that A + B is symmetri...
 1.5.25: (a) Show that ir A is an upper triangular matrix. then AT is lower ...
 1.5.26: If A is a skew symmetric m.atrix. whal Iype of malrix is AT? Justif...
 1.5.27: Show that if A is skew sym ~t ri c, then the elements on lhe main d...
 1.5.28: Show that if A is skew symllletric, the n A' is skew sym metric for...
 1.5.29: Show 1hat if A is an It x II ma\Jix. then A = S + K . where S is sy...
 1.5.30: Let Find the matrices Sand K described in Exercise 29.
 1.5.31: Show that the m:l1rix A = [! !] is singular.
 1.5.32: IfD = [~ o  2 o
 1.5.33: Find the inverse of each of the following matrices: (a) A = [! ;] (...
 1.5.34: If A is a nonsingular matrix whose inverse is [~ :l fi nd A.
 1.5.35: If and fi nd (AB) I
 1.5.36: Suppose that 37. AI =[: ~l Solve the linear system Ax = h for each ...
 1.5.37: The linear system ACx nonsingular with
 1.5.38: The linear system ~ = b is such that A is nonsingular wilh Find the...
 1.5.39: The linear system AT x = h is such that A is nonsingular with A l= ...
 1.5.40: The linear system C T Ax = b is such that A and C are nonsingular. ...
 1.5.41: Consider the linear syMem Ax = h. where A is the mao trix defined i...
 1.5.42: Find t .... "O 2 x 2 singular matrices whose sum is nonsin gular.
 1.5.43: Find twO 2 x 2 nonsUlgular matrices whose sum ii sin gular.
 1.5.44: Pro\'e Corollary I. L
 1.5.45: Pro\'e Theorem 1.7.
 1.5.46: Prove Ihal if one row (column) of the n X II matrix A con sists e n...
 1.5.47: Prove: If A is a diagonal illlitrix with nonzero di agonal el11ries...
 1.5.48: Lo< A = [~ o 3 o
 1.5.49: For an /I x /I diagonal matrix A whose diagonal entries arc lIll' l...
 1.5.50: Show Ihat if A B 1J =c.
 1.5.51: Show that if A is nonsingular and A H = 0 for an /I x /I nlatrix H....
 1.5.52: Let A = [:. : l Show that A i~ nonsingular if and only if lid  be ...
 1.5.53: Consider the homogeneous system Ax = O. where A is /I X 11. If A is...
 1.5.54: Pro\c that if A is symmetric and non ingular. then A I IS symmetric
 1.5.55: Formulate the methoo for adding panitioned matrices. 9nd verify you...
 1.5.56: Let A and lJ be the following matrices: AJ~ 1 3 4 1] 2 3  I 3 2 1 ...
 1.5.57: What type of matrix is a linear combination o f symmetric matrices?...
 1.5.58: Whm type of matrix is a linear combination o f scalar matrices? Jus...
 1.5.59: The matrix form of the recursion relmion 11 0 = O. Ifl = I. II ~ = ...
 1.5.60: The matrix form of the recursion relution is written as where [ ""J...
 1.5.61: For the software you are Ilsing. determine the command(s) or proced...
 1.5.62: Most software for linear algebra ha~ specific commands for extracti...
 1.5.63: Determine the command for computing the inverse of a matrix in the ...
 1.5.64: If B is the inverse of II x /I matrix A. then Definition 1.10 guara...
 1.5.65: In Section 1.1 we studied the method of elimination for solving lin...
 1.5.66: For the software you are using. determine the command for obtaining...
 1.5.67: Experiment with your software to determine the b~hav ior of the mat...
Solutions for Chapter 1.5: Special Types of Matrices and Partitioned Matrices
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 1.5: Special Types of Matrices and Partitioned Matrices
Get Full SolutionsSince 67 problems in chapter 1.5: Special Types of Matrices and Partitioned Matrices have been answered, more than 9329 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. Chapter 1.5: Special Types of Matrices and Partitioned Matrices includes 67 full stepbystep solutions.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.