FAST-NU Entry Test Advanced Mathematics Flashcards

24 sample cards from the Advanced Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What four axioms must (G, *) satisfy to be a group?

   Closure, Associativity, existence of an Identity element e (a*e = a), and existence of an Inverse for each element (a*a⁻¹ = e).

2. What additional property makes a group abelian (commutative)?

   A group is abelian if a*b = b*a for all a, b ∈ G — the operation is commutative.

3. Distinguish a semigroup, a monoid, and a group.

   Semigroup: closed + associative. Monoid: semigroup + identity. Group: monoid + every element has an inverse.

4. Define a square matrix, a diagonal matrix, and a scalar matrix.

   Square: equal rows and columns (n×n). Diagonal: square with all off-diagonal entries 0. Scalar: a diagonal matrix with all diagonal entries equal.

5. Define an identity matrix and a null (zero) matrix.

   Identity I: diagonal matrix with 1s on the diagonal (AI = IA = A). Null matrix: all entries are 0 (A + 0 = A).

6. What is a symmetric matrix versus a skew-symmetric matrix?

   Symmetric: Aᵀ = A (aᵢⱼ = aⱼᵢ). Skew-symmetric: Aᵀ = −A (aᵢⱼ = −aⱼᵢ), forcing zeros on the main diagonal.

7. State the rule for matrix multiplication conformability and the order of the product.

   To compute A·B, columns of A must equal rows of B. If A is m×n and B is n×p, AB is m×p. Matrix multiplication is generally non-commutative (AB ≠ BA).

8. Define the transpose of a matrix and state (AB)ᵀ.

   Transpose Aᵀ swaps rows and columns. (AB)ᵀ = BᵀAᵀ (order reverses); also (Aᵀ)ᵀ = A and (A+B)ᵀ = Aᵀ+Bᵀ.

9. How is the determinant of a 2×2 matrix [[a,b],[c,d]] computed?

   det = ad − bc. For 2×2 matrices the determinant is the cross-product difference of the diagonals.

10. How do you expand the determinant of a 3×3 matrix by cofactors along the first row?

    For [[a,b,c],[d,e,f],[g,h,i]]: det = a(ei−fh) − b(di−fg) + c(dh−eg), using alternating + − + signs with 2×2 minors.

11. State two key properties of determinants under row operations.

    Swapping two rows multiplies det by −1; multiplying a row by k multiplies det by k. If two rows are identical or a row is all zeros, det = 0.

12. What is det(AB) and det(Aᵀ) in terms of det(A) and det(B)?

    det(AB) = det(A)·det(B), and det(Aᵀ) = det(A). Also det(kA) = kⁿ·det(A) for an n×n matrix.

13. Define a singular matrix and a non-singular matrix.

    Singular: det = 0 (no inverse exists). Non-singular: det ≠ 0 (invertible). Only non-singular square matrices have inverses.

14. State the adjoint formula for the inverse of a matrix.

    A⁻¹ = (1/det A)·adj(A), where adj(A) is the transpose of the cofactor matrix. Requires det A ≠ 0.

15. Give the quick formula for the inverse of a 2×2 matrix [[a,b],[c,d]].

    A⁻¹ = 1/(ad−bc) · [[d, −b],[−c, a]], provided ad − bc ≠ 0.

16. State the property (AB)⁻¹ and the relation between A⁻¹ and A.

    (AB)⁻¹ = B⁻¹A⁻¹ (order reverses). Also A·A⁻¹ = A⁻¹·A = I and det(A⁻¹) = 1/det(A).

17. State Cramer's rule for solving a linear system AX = B.

    For a non-singular system, each variable xᵢ = det(Aᵢ)/det(A), where Aᵢ is A with its i-th column replaced by the constants B.

18. Describe the matrix-inverse method for solving AX = B.

    If A is non-singular, the unique solution is X = A⁻¹B. Compute A⁻¹ and multiply by the constant vector B.

19. How does the determinant of the coefficient matrix indicate the nature of a linear system's solutions?

    If det A ≠ 0: unique solution. If det A = 0: the system has either no solution (inconsistent) or infinitely many solutions (dependent).

20. What is the discriminant of a quadratic ax² + bx + c = 0, and what does it determine?

    Discriminant D = b² − 4ac. It determines the nature (real/complex, distinct/equal) of the roots without solving the equation.

21. Give the nature of roots for each case of the discriminant D = b² − 4ac.

    D > 0: two real distinct roots. D = 0: two real equal (repeated) roots. D < 0: two complex conjugate roots. D a perfect square (D>0): rational distinct roots.

22. State the quadratic formula for the roots of ax² + bx + c = 0.

    x = [−b ± √(b²−4ac)] / (2a), giving the two roots of the quadratic equation.

23. For ax² + bx + c = 0 with roots α and β, what are the sum and product of the roots?

    Sum α + β = −b/a; Product αβ = c/a. (Vieta's formulas for a quadratic.)

24. How do you form a quadratic equation given the sum S and product P of its roots?

    The equation is x² − Sx + P = 0, i.e. x² − (sum of roots)x + (product of roots) = 0.

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Advanced Mathematics flashcards FAQ