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Solutions for Chapter 4-7: Identity and Inverse Matrices

Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition | ISBN: 9780078738302 | Authors: McGraw-Hill Education

Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition

ISBN: 9780078738302

Algebra 2, Student Edition (MERRILL ALGEBRA 2) | 1st Edition | ISBN: 9780078738302 | Authors: McGraw-Hill Education

Solutions for Chapter 4-7: Identity and Inverse Matrices

Solutions for Chapter 4-7
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Textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2)
Edition: 1
Author: McGraw-Hill Education
ISBN: 9780078738302

Chapter 4-7: Identity and Inverse Matrices includes 75 full step-by-step solutions. Since 75 problems in chapter 4-7: Identity and Inverse Matrices have been answered, more than 56532 students have viewed full step-by-step 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: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302.

Key Math Terms and definitions covered in this textbook
  • Affine transformation

    Tv = Av + Vo = linear transformation plus shift.

  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • Back substitution.

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

  • Cholesky factorization

    A = CTC = (L.J]))(L.J]))T for positive definite A.

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = 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.

  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Inverse matrix A-I.

    Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Multiplication Ax

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

  • Network.

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

  • 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.

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • 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.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Reflection matrix (Householder) Q = I -2uuT.

    Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

  • Symmetric factorizations A = LDLT and A = QAQT.

    Signs in A = signs in D.

  • Vector addition.

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

  • Wavelets Wjk(t).

    Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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