3.1-3.2 Matrix arithmetic and inverses Quiz: Pearson Edexcel Further Maths, Unit 3

The 20 questions

1. What condition must hold for the product AB of matrices A and B to be defined?

   • A and B have the same number of rows
   • A and B must both be square matrices
   • The number of columns of A equals the number of rows of B
   • A and B have the same number of columns

2. What is the size of the product of a 2 x 3 matrix and a 3 x 4 matrix?

   • 4 x 2
   • 2 x 4
   • 2 x 3
   • 3 x 3

3. Which matrix is the 2 x 2 identity matrix?

   • [[0, 1], [1, 0]]
   • [[1, 1], [1, 1]]
   • [[0, 0], [0, 0]]
   • [[1, 0], [0, 1]]

4. Is matrix multiplication commutative in general?

   • No, but only when one matrix is the zero matrix
   • Yes, but only when both matrices are singular
   • No, AB and BA can be different matrices
   • Yes, AB always equals BA

5. What is the effect of multiplying a matrix A by the scalar 3?

   • A is replaced by its transpose
   • 3 is added to every entry of A
   • Every entry of A is multiplied by 3
   • Only the diagonal entries of A are multiplied by 3

6. For a 2 x 2 matrix [[a, b], [c, d]], when does its inverse exist?

   • When a equals d
   • When b equals c
   • When a + d is not zero
   • When ad - bc is not zero

7. Which expression correctly gives the inverse of the product AB of invertible matrices A and B?

   • A^(-1) B^(-1)
   • B^(-1) A^(-1)
   • B A
   • (A + B)^(-1)

8. For A = [[1, 2], [3, 4]], what is the (1,2) entry of A squared?

   • 22
   • 8
   • 10
   • 14

9. For A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], what is 2A - B?

   • [[2, 3], [5, 8]]
   • [[2, 5], [7, 8]]
   • [[1, 3], [5, 4]]
   • [[2, 4], [6, 8]]

10. For A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], what is AB?

    • [[0, 1], [1, 0]]
    • [[2, 1], [4, 3]]
    • [[3, 4], [1, 2]]
    • [[1, 2], [3, 4]]

11. What is the inverse of the matrix [[1, 2], [3, 4]]?

    • [[2, -1], [-3/2, 1/2]]
    • [[-2, 1], [3/2, 1/2]]
    • [[4, -2], [-3, 1]]
    • [[-2, 1], [3/2, -1/2]]

12. Which of these matrices has no inverse?

    • [[2, 4], [1, 2]]
    • [[1, 0], [0, 1]]
    • [[1, 2], [3, 5]]
    • [[2, 1], [1, 2]]

13. A matrix product is formed from a 3 x 2 matrix and a 2 x 3 matrix. What is its size?

    • 3 x 3
    • 2 x 3
    • 3 x 2
    • 2 x 2

14. For A = [[1, 2], [3, 4]], solve Ax = b where b = [5, 11] and x is a column vector. What is x?

    • [-1, 3]
    • [2, 1]
    • [1, 2]
    • [5, 11]

15. If A^(-1) = [[2, 1], [1, 1]], what is A?

    • [[2, -1], [-1, 1]]
    • [[-2, 1], [1, -1]]
    • [[1, -1], [-1, 2]]
    • [[1, 1], [1, 2]]

16. For M = [[1, 1], [0, 1]], what is M^3?

    • [[1, 3], [0, 1]]
    • [[1, 2], [0, 1]]
    • [[1, 1], [0, 1]]
    • [[3, 3], [0, 3]]

17. For the matrix [[1, 2], [3, k]] to be singular, what value must k take?

    • 3
    • 2
    • -6
    • 6

18. For A = [[1, 2], [3, 4]] and X satisfying MX = N with M = A and N = I, what is X?

    • [[4, -2], [-3, 1]]
    • [[-2, 1], [3/2, -1/2]]
    • [[-2, 1], [3/2, 1/2]]
    • [[1, 2], [3, 4]]

19. If A is singular and B is invertible, which statement about AB is true?

    • AB is invertible whenever B is square
    • AB is invertible
    • AB is singular
    • AB equals the identity matrix

20. For A = [[1, 2], [3, 4]], what is det(A^2)?

    • 4
    • 2
    • -4
    • -2

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