8 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-22.

In Problems 1–20, fill in the blanks or answer true/false.

  1. A matrix A = (aij)4×3 such that aij = i + j is given by _____.
  2. If A is a 4 × 7 matrix and B is a 7 × 3 matrix, then the size of AB is _____.
  3. If A = and B = (3 4), then AB = _____ and BA = _____.
  4. If A = , then A−1 = _____.
  5. If A and B are n × n nonsingular matrices, then A + B is necessarily nonsingular. _____
  6. If A is a nonsingular matrix for which AB = AC, then B = C. _____
  7. If A is a 3 × 3 matrix such that det A = 5, then det(A) = _____ and det(−AT) =_____.
  8. If det A = 6 and det B = 2, then det AB−1 = _____.
  9. If A and B are n × n matrices whose corresponding entries in the third column are the same, then det(A − B) = _____.
  10. Suppose A is a 3 × 3 matrix such that det A = 2. If B = 10A and C = −B−1, then det C = _____.
  11. Let A be an n × n matrix. The eigenvalues of A are the nonzero solutions of det(A − λI) = 0. _____
  12. A nonzero scalar multiple of an eigenvector is also an eigenvector corresponding to the same eigenvalue. _____
  13. An n × 1 column vector K with all zero entries is never an eigenvector of an n × n matrix A. _____
  14. Let A be an n × n matrix with real entries. If λ is a complex eigenvalue, then is also an eigenvalue of A. _____
  15. An n × n matrix A always possesses n linearly independent eigenvectors. _____
  16. The augmented matrix is in reduced row-echelon form. _____
  17. If a 3 × 3 matrix A is diagonalizable, then it possesses three linearly independent eigenvectors. _____
  18. The only matrices that are orthogonally diagonalizable are symmetric matrices. _____
  19. The symmetric matrix A = is orthogonal. _____
  20. The eigenvalues of a symmetric matrix with real entries are always real numbers. _____
  21. An n × n matrix B is symmetric if BT = B, and an n × n matrix C is skew-symmetric if CT = −C. By noting the identity 2A = A + AT + A − AT, show that any n × n matrix A can be written as the sum of a symmetric matrix and a skew-symmetric matrix.
  22. Show that there exists no 2 × 2 matrix with real entries such that A2 = .
  23. An n × n matrix A is said to be nilpotent if, for some positive integer m, Am = 0. Find a 2 × 2 nilpotent matrix A ≠ 0.
    1. Two n × n matrices A and B are said to anticommute if AB = −BA. Show that each of the Pauli spin matrices

      where i2 = −1, anticommutes with the others. Wolfgang Ernst Pauli (1900–1958) was an Austrian-born theoretical physicist and one the pioneers of quantum physics. Pauli received the Nobel Prize in Physics in 1945.

    2. The matrix C = AB − BA is said to be the commutator of the n × n matrices A and B . Find the commutators of σx and σy, σy and σz, and σz and σx.

In Problems 25 and 26, solve the given system of equations by Gauss–Jordan elimination.

  3. Without expanding, show that .
  4. Show that = 0 is the equation of a parabola passing through the three points (1, 2), (2, 3), and (3, 5).

In Problems 29 and 30, evaluate the determinant of the given matrix by inspection.

In Problems 31 and 32, without solving, state whether the given homogeneous system has only the trivial solution or has infinitely many solutions.

In Problems 33 and 34, use Gauss–Jordan elimination to balance the given chemical equation.

  1. I2 + HNO3 → HIO3 + NO2 + H2O
  2. Ca + H3PO4 → Ca3P2O8 + H2

In Problems 35 and 36, solve the given system of equations by Cramer’s rule.

  3. Use Cramer’s rule to solve the system

    for x and y.

    1. Set up the system of equations for the currents in the branches of the network given in FIGURE 8.R.1.
    2. Use Cramer’s rule to show that

      An electric network consists of a battery E and three resistors R subscript 1, R subscript 2, R subscript 3 in parallel. The current i subscript 1 flows out of the battery. After branching out from the nodes the current flowing through the resistors R subscript 1, R subscript 2 and R subscript 3 are i subscript 1, i subscript 2, and i subscript 3 respectively.

      FIGURE 8.R.1 Network in Problem 38

  5. Solve the system

    by writing it as a matrix equation and finding the inverse of the coefficient matrix.

  6. Use the inverse of the matrix A to solve the system AX = B, where

    and the vector B is given by (a) (b) .

In Problems 41–46, find the eigenvalues and corresponding eigenvectors of the given matrix.

  7. Supply a first column so that the matrix is orthogonal:

  8. Consider the symmetric matrix A = .
    1. Find matrices P and P−1 that orthogonally diagonalize the matrix A.
    2. Find the diagonal matrix D by actually carrying out the multiplication P−1AP.
  9. Identify the conic section x2 + 3xy + y2 = 1.
  10. Consider the following population data:

    A table has 6 columns and 1 row. The column headings from left to right are Year, 1890, 1900, 1910, 1920, 1930. The row entries are as follows. Row 1: Population (in millions). 1890, 63. 1900, 76. 1910, 92. 1920, 106. 1930, 123.

    The actual population in 1940 was 132 million. Compare this amount with the population predicted from the least squares line for the given data.

In Problems 51 and 52, use the matrix A = to encode the given message. Use the correspondence (1) of Section 8.14.

  1. SATELLITE LAUNCHED ON FRI
  2. SEC AGNT ARRVS TUES AM

In Problems 53 and 54, use the matrix A = to decode the given message. Use the correspondence (1) in Section 8.14.

  3. Decode the following messages using the parity check code.

    (a) (1 1 0 0 1 1)

    (b) (0 1 1 0 1 1 1 0)

  4. Encode the word (1 0 0 1) using the Hamming (7, 4) code.

In Problems 57 and 58, solve the given system of equations using LU-factorization.

  1. The system in Problem 26
  2. The system in Problem 36
  3. Find the least squares line for the data (1, −2), (2, 0), (3, 5), (4, −1).
  4. Find the least squares parabola for the data given in Problem 59.