Mathematical methods for economic theory

Martin J. Osborne
Previous section Previous page Contents Text Exercises Search Next page Next section

1.2 Exercises on matrices: determinant, inverse, and rank

  1. Let
    A = 
    left parenthesis 4 −1 right parenthesis
    6 9
    and
    B = 
    left parenthesis 0 3 right parenthesis
    3 −2
    Find (i) A + B, (ii) 2A − B, (iii) AB, (iv) BA, and (v) A' (the transpose of A).

    Solution

    1. left parenthesis 4 2 right parenthesis
      9 7
    2. left parenthesis 8 −5 right parenthesis
      9 20
    3. left parenthesis −3 14 right parenthesis
      27 0
    4. left parenthesis 18 27 right parenthesis
      0 −21
    5. left parenthesis 4 6 right parenthesis
      −1 9

  2. Let
    A = 
    left parenthesis 4 −1 right parenthesis
    6 9
    2 3
    and
    B = 
    left parenthesis 0 3 right parenthesis
    3 −2
    (i) Is AB defined? If so, find it. (ii) Is BA defined? If so, find it.

    Solution

    1. Yes;
      left parenthesis −3 14 right parenthesis
      27 0
      9 0
    2. No

  3. Given u' = (5,2,3), find u'·u (the scalar product, or inner product).

    Solution

    u'·u = 38.

  4. You buy n items in the quantities q1,...,qn at the prices p1,...,pn. Express your expenditure using (i) Σ notation, (ii) vector notation.

    Solution

    (i) ∑n
    i=1    piqi; (ii) p·q' where p = (p1, ..., pn) and q = (q1, ..., qn).

  5. Let x' = (x1, x2) and let
    A = 
    left parenthesis a11 a12 right parenthesis
    a21 a22
    Find x'Ax.

    Solution

    a11x2
    1     + (a12 + a21)x1x2 + a22x2
    2    .

  6. Find the determinants of the matrices A and B in Problem 1.

    Solution

    |A| = 42. |B| = −9.

  7. Let A be an n × n matrix in which all the elements are zero except those on or below the main diagonal. Find an expression for the determinant of A.

    Solution

    Calculating the determinant by expanding along its first row, and then expanding each of the determinants of the smaller matrices along their first rows, we find that |A| = a11a22···ann. (That is, the determinant of the matrix is the same as the determinant of the matrix that differs from A only in that all the elements below the main diagonal, as well as all the elements above the main diagonal, are zero.)

  8. Find the inverse of the matrix A in Problem 1, and verify that it is indeed the inverse.

    Solution

    A−1 = (1/42)
    left parenthesis 9 1 right parenthesis
    −6 4
    		; A−1A = I.

    1. Find the determinants of the following matrices.
      A = 
      left parenthesis 8 1 3 right parenthesis
      4 0 1
      6 0 3
      B = 
      left parenthesis a b c right parenthesis
      b c a
      c a b

      Solution

      |A| = −6; |B| = 3abc−a3 − b3 − c3.

    2. Are either of these matrices nonsingular?

      Solution

      A: yes; B: if and only if 3abc ≠ a3 + b3 + c3.

    3. Find the inverse of the matrix A.

      Solution

      A−1 = 
      left parenthesis 0 1/2 −1/6 right parenthesis
      1 −1 −2/3
      0 −1 2/3
  10. Find the rank of the following matrix:
    left parenthesis −1 0 2 1 right parenthesis
    −2 2 4 2
    −3 1 6 3
    		.

    Solution

    Every 3 × 3 submatrix has determinant zero, but the 2 × 2 matrix obtained by deleting the third row and second and third columns has determinant −2 ≠ 0, so the rank of the matrix is 2.