C7-C8 Solving three simultaneous equations with matrices Quiz: AQA Further Maths, Unit 2

The 20 questions

1. If A is non-singular and AX = B, what is X?

   • A + B
   • A B^-1
   • A^-1 B
   • B A^-1

2. Geometrically, three planes with a unique common solution meet in what?

   • A plane
   • A line
   • A single point
   • No point at all

3. If the coefficient matrix of a 3 x 3 system is singular, what can happen?

   • No solution or infinitely many solutions
   • Exactly three solutions, one for each of the three planes in the system
   • Exactly two solutions, since two of the three planes always meet in a line
   • Exactly one solution always, since three planes always meet at one point

4. Solve x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2.

   • x = 1, y = 3, z = 2
   • x = 1, y = 2, z = 3
   • x = 2, y = 1, z = 3
   • x = 3, y = 2, z = 1

5. What is the determinant of [[1, 1, 1], [2, -1, 1], [1, 2, -1]]?

   • 5
   • 1
   • 7
   • -7

6. The planes x + y + z = 1 and x + y + z = 2 together with x + 2y + 3z = 0 are arranged how?

   • A single common point, since three planes usually meet at one point
   • A line of common points, since any two planes in space meet along a line
   • No common point, because the first two planes are parallel
   • Every point of space, since the three equations are all satisfied by zero

7. The planes x + 2y + 3z = 6, 2x + 4y + 6z = 12 and x + y + z = 3 are arranged how?

   • They meet in a unique point
   • They meet in a line of solutions
   • They have no common point
   • They are a single plane

8. For which k does x + y + z = 1, x + 2y + 3z = 2, x + 3y + kz = 3 have a unique solution?

   • All k except k = 5
   • k = 0 only, since the coefficient matrix is then the identity matrix
   • All real k, since the system has a unique solution for every value of k
   • k = 5 only, since then the determinant of the coefficient matrix is zero

9. For which k is [[1, 2, 3], [2, k, 6], [1, 1, 1]] singular?

   • k = -4
   • k = 4
   • k = 2
   • k = 6

10. Solve x + y + z = 3, x - y = 0, z = 1.

    • x = 1, y = 0, z = 2
    • x = 1, y = 1, z = 1
    • x = 2, y = 2, z = -1
    • x = 0, y = 2, z = 1

11. Solve x + 2y + 3z = 14, y + z = 4, z = 2 by back substitution.

    • x = 6, y = 2, z = 2
    • x = 2, y = 4, z = 2
    • x = 4, y = 2, z = 2
    • x = 4, y = 4, z = 2

12. For which k is [[1, 1, 1], [1, k, 1], [1, 1, 1]] singular?

    • All real k
    • k = 0 only
    • No real k
    • k = 1 only

13. What is the determinant of the coefficient matrix for x - y + z = 1, 2x + y = 3, x + z = 2?

    • -2
    • 2
    • 1
    • 4

14. Which statement about AX = B is always true when A is non-singular?

    • There are infinitely many solutions
    • X = A^-1 B is the unique solution
    • The system has no solution
    • X = B A^-1 is always the solution

15. Solve x + y + z = 6, x - y + z = 2, x + y - z = 0 for z.

    • z = 2
    • z = 6
    • z = 3
    • z = 1

16. For A = diag(1, 1, 2) and B = (1, 2, 3), solve AX = B.

    • x = 1/2, y = 1, z = 3/2
    • x = 1, y = 2, z = 3
    • x = 1, y = 2, z = 6
    • x = 1, y = 2, z = 3/2

17. What is det [[1, 0, 0], [0, 1, 0], [0, 0, 2]]?

    • 1
    • 0
    • 3
    • 2

18. What is the inverse of the diagonal matrix diag(2, 1, 1)?

    • diag(-2, -1, -1)
    • diag(1/2, 1, 1)
    • diag(1/2, 0, 0)
    • diag(2, 1, 1)

19. Which system has a unique solution?

    • x + y + z = 0 with no other equations, so there are infinitely many solutions
    • x + y + z = 0, x - y = 0, z = 0, with x = y = z = 0
    • x + y + z = 0 and x - y = 0 only, which leaves a whole line of solutions
    • x + y = 1, x + y = 2, z = 0, which describes parallel planes with no solution

20. Solve 2x = 4, y - z = 1 and y + z = 5.

    • x = 4, y = 3, z = 2
    • x = 2, y = 3, z = 2
    • x = 2, y = 2, z = 3
    • x = 2, y = 3, z = 4

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