 2.4.1E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.2E: In Exercises, assume that the matrices are partitioned conformably ...
 2.4.3E: In Exercises, assume that the matrices are partitioned conformably ...
 2.4.4E: In Exercises, assume that the matrices are partitioned conformably ...
 2.4.5E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.6E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.7E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.8E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.9E: In Exercises, assume that the matrices are partitioned conformably ...
 2.4.10E: The inverse of is . Find X, Y, and Z.
 2.4.11E: In Exercises 11 and 12, mark each statement True or False. Justify ...
 2.4.12E: In Exercises, mark each statement True or False. Justify each answe...
 2.4.13E: Let where B and C are square. Show that A is invertible if and only...
 2.4.14E: Show that the block upper triangular matrix A in Example 5 is inver...
 2.4.15E: Let If A11 is invertible, then the matrix is called the Schur compl...
 2.4.16E: Suppose the block matrix A on the left side of (7) is invertible an...
 2.4.17E: When a deep space probe is launched, corrections may be necessary t...
 2.4.18E: Let X be an m × n data matrix such that XT X is invertible, and let...
 2.4.19E: In the study of engineering control of physical systems, a standard...
 2.4.20E: In the study of engineering control of physical systems, a standard...
 2.4.21E: a. Verify that A2 = I when .b. Use partitioned matrices to show tha...
 2.4.22E: Generalize the idea of Exercise [not 21(b)] by constructing a 5 × 5...
 2.4.23E: Use partitioned matrices to prove by induction that the product of ...
 2.4.24E: Use partitioned matrices to prove by induction that for n = 2, 3,…,...
 2.4.25E: Without using row reduction, find the inverse of
 2.4.26E: [M] For block operations, it may be necessary to access or enter su...
 2.4.27E: [M] Suppose memory or size restrictions prevent a matrix program fr...
Solutions for Chapter 2.4: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 2.4
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Chapter 2.4 includes 27 full stepbystep solutions. Since 27 problems in chapter 2.4 have been answered, more than 43860 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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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.

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

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

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.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.