Q5. How many different orders can a matrix with 12 elements have? List them.
12 = 1 × 12 = 2 × 6 = 3 × 4, each either way round: 1 × 12, 12 × 1, 2 × 6, 6 × 2, 3 × 4, 4 × 3. Six orders.
Q6. Construct the 3 × 3 matrix with aij = the larger of i and j, and the one with aij = (−1)i + j.
Larger of i and j: rows 1, 2, 3; 2, 2, 3; 3, 3, 3. (−1)i + j: rows 1, −1, 1; −1, 1, −1; 1, −1, 1, a chessboard of signs.
Q7. A matrix has 7 elements. What can you say about its shape?
7 is prime, so the only orders are 1 × 7 (a single row) and 7 × 1 (a single column).
Q8. Write the triangle with vertices P(2, 1), Q(−1, 4) and R(0, −3) as a 2 × 3 matrix with one column per vertex. What is the element in row 2, column 3, and what does it mean?
The columns are (2, 1), (−1, 4) and (0, −3), so row 1 is 2, −1, 0 and row 2 is 1, 4, −3. The element a23 = −3 is the y-coordinate of R.
Q9. How many 2 × 2 matrices have every entry from {0, 1, 2}? How many of these have all four entries different?
4 entries with 3 choices each: 34 = 81. All four different is impossible, because only 3 values are available: 0.