NUST NET Mathematics Flashcards

24 sample cards from the Mathematics deck

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

1. Is matrix multiplication commutative? State the correct property.

   No; in general AB ≠ BA. Matrix multiplication is associative and distributive but not commutative.

2. What is the transpose of a matrix, and what is (AB)ᵀ?

   The transpose Aᵀ is formed by interchanging rows and columns. (AB)ᵀ = BᵀAᵀ.

3. How do you compute the determinant of a 2×2 matrix [[a, b],[c, d]]?

   det = ad − bc.

4. How do you compute the determinant of a 3×3 matrix by cofactor expansion along the first row?

   det = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁).

5. What happens to a determinant if two rows (or columns) are interchanged?

   The determinant changes sign (is multiplied by −1).

6. What is the value of a determinant if two rows or columns are identical or proportional?

   The determinant is zero.

7. State the relationship between det(AB), det(A), and det(B), and det(Aᵀ).

   det(AB) = det(A)·det(B), and det(Aᵀ) = det(A).

8. What is a singular matrix and a non-singular matrix?

   Singular: det(A) = 0 (no inverse). Non-singular: det(A) ≠ 0 (invertible).

9. What is the formula for the inverse of a matrix A in terms of its adjoint?

   A⁻¹ = (1/det A) · adj(A), valid when det A ≠ 0.

10. How do you find the inverse of a 2×2 matrix [[a, b],[c, d]]?

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

11. What is the adjoint (adjugate) of a matrix?

    The transpose of the matrix of cofactors of A.

12. State the property (AB)⁻¹ and (A⁻¹)⁻¹.

    (AB)⁻¹ = B⁻¹A⁻¹ and (A⁻¹)⁻¹ = A.

13. State Cramer's Rule for solving a system of linear equations.

    For AX = B with det A ≠ 0, each variable xᵢ = det(Aᵢ)/det(A), where Aᵢ is A with its i-th column replaced by B.

14. In the matrix method, how is the solution of AX = B found when A is non-singular?

    X = A⁻¹B, computed by multiplying the inverse of the coefficient matrix by the constant matrix.

15. When does a system of linear equations have a unique solution, no solution, or infinitely many solutions (in terms of the determinant)?

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

16. What is the quadratic formula for ax² + bx + c = 0?

    x = [−b ± √(b² − 4ac)] / (2a).

17. Name the four standard methods of solving a quadratic equation.

    Factorization, completing the square, the quadratic formula, and graphical method.

18. What is the discriminant of a quadratic equation, and what symbol denotes it?

    D = b² − 4ac (the expression under the square root in the quadratic formula).

19. How does the discriminant determine the nature of the roots of ax² + bx + c = 0?

    D > 0: real and distinct roots. D = 0: real and equal roots. D < 0: complex conjugate (imaginary) roots.

20. When are the roots of a quadratic real and rational versus real and irrational?

    If D = b²−4ac is a perfect square (and a,b,c rational): rational roots. If D > 0 but not a perfect square: irrational (conjugate surd) roots.

21. 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.

22. How do you form a quadratic equation given its roots α and β?

    x² − (α + β)x + αβ = 0, i.e., x² − (sum of roots)x + (product of roots) = 0.

23. How is an equation of the form ax⁴ + bx² + c = 0 reduced to a quadratic?

    Substitute y = x², giving ay² + by + c = 0; solve for y, then back-substitute x = ±√y.

24. How can an equation like x + √x − 6 = 0 be reduced to quadratic form?

    Let y = √x (y ≥ 0), giving y² + y − 6 = 0; solve for y, then x = y² (rejecting negative y).

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