FAD1015 Tutorial 13 — Matrices

Questions

  1. State the type of the following matrices: (a) (15 0; 0 22)
    (b) (0 11; 77 0)
    (c) (0 0; 0 0)
    (d) (44 11; 0 0)
    (e) (2 0; 8 7)
    (f) (0 0; 0 -3)
    (g) (5 1; 8 7)
    (h) (G K; 5 23; s 1; 0 8)
    (i) (-4 0 2; 3 -2 0; h 3 19)

  2. Find a general 3 x 3 matrix that satisfies the stated condition: (a) a_ij = 0 if i ≠ j (diagonal matrix) (b) a_ij = 0 if i < j (upper triangular matrix) (c) a_ij = 0 if i > j (lower triangular matrix) (d) a_ij = 0 if |i - j| > 1 (tridiagonal matrix)

  3. Given A = [2 3; -1 0] B = [1 -1; 4 2] C = [1 4; -3 2; -1] D = [1 3; -1 2; 3 4]

    Find: (a) A + B (b) 2C + 3B (c) 4C - 2D (d) 6D (e) M(A + B), M is a scalar (f) BD (g) D^T C^T (h) (AB^T)^T (i) IA

  4. State whether the following property of matrix is TRUE or FALSE: If A, B are matrices of the same order, C and D are (m × n) and (n × p) matrices respectively, 0 is a zero matrix, k and m is a scalar, (a) A + 0 = 0 + A is true from the associative property. (b) A + (-A) = (-A) + A. (c) mA + kA = A(m + k). (d) CD = DC. (e) CD = 0 imply that C = 0, D = 0.

  5. Arrange the matrices (5 2; 4 1), (1), and (5; 4) such that the product of the three matrices is a 2 × 2 matrix. Find this product.

  6. Suppose matrix G = [a c; b d] and G^2 + pG + qI = 0, where I is an identity matrix and 0 is the zero matrix. Find p and q.

  7. If A = [u b; v w], B = [x d; y z], show that (AB)^T = B^T A^T.

  8. Evaluate the determinant of matrix A = [-2 0 -3; 1 0 -4] using cofactor expansion.

  9. By using the properties of determinants, state whether the following statements are TRUE or FALSE. Give reasons for your answer. (a) |5 4 2; 9 7 5; 3 5 9| = |2 4 7; 9 9 1; 2 7 2| (b) |8 6 4; 9 9 1; 9 2 7| = |1 6 4; 9 9 1; 9 2 7| (c) |0 7 5; 4 4 5; 5 4 4| = - |5 7 0; 4 4 5; 5 4 4| (d) |5 1 5; 7 5 7; 6 4 2| = 0 (e) |0 7 5; 0 0 1; 3 7 5| = 3 × 7 × 1

Key Topics Covered

  • Matrix types (diagonal, triangular, identity, zero)
  • Matrix operations (addition, subtraction, scalar multiplication)
  • Matrix multiplication
  • Transpose of matrices
  • Determinants and their properties
  • Matrix equations

Links

  • FAD1015 - Mathematics III
  • Matrices
  • Matrix Types
  • Matrix Operations
  • Determinants