How often Determinants and Cramer's rule is asked

Where it was asked

• May/Jun 201225 marks
• Oct/Nov 201115 marks
• May/Jun 201118 marks

The questions

1. May/Jun 2012, Q2.15 marks

   Consider the 3x3 matrix A whose rows are 4, -1, 5 in the first row; 6, 2, 1 in the second row; and 0, 2, 1 in the third row. Write down this matrix A and calculate its determinant.

2. May/Jun 2012, Q2.24 marks

   Consider the 3x3 matrix B whose first row is cos(theta), sin(theta), 0; whose second row is -sin(theta), cos(theta), 0; and whose third row is 0, 0, 1. Show that B is invertible for every value of theta. (Hint: show that det(B) is never equal to 0 for any value of theta.)

3. May/Jun 2012, Q2.39 marks

   Let C be the 3x3 matrix with first row 2, -1, 3; second row 1, 2, 4; and third row 5, -3, 6. Show that the determinant of C equals the determinant of the transpose of C, that is, show that det(C) = det(C^T).

4. May/Jun 2012, Q2.47 marks

   Using Cramer's rule, solve the following system of two equations in x and y: 2x - y = 4, and 3x + 2y = 13.

5. Oct/Nov 2011, Q3.17 marks

   Given the matrix A = [[2,1,3],[0,-1,2],[3,4,-1]], calculate det(A) by expanding specifically along the second column, showing every step of the working. Note that no marks will be given if any other expansion method or technique is used.