FSc Pre-Engineering 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. Define the identity matrix and the null (zero) matrix.

   Identity matrix $I$: a diagonal matrix with all diagonal entries $1$; e.g. $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Null matrix: every entry is $0$.

2. Define symmetric and skew-symmetric matrices.

   Symmetric: $A^{t} = A$ (i.e. $a_{ij} = a_{ji}$). Skew-symmetric: $A^{t} = -A$ (i.e. $a_{ij} = -a_{ji}$, so diagonal entries are $0$).

3. When can two matrices be added, and how is addition performed?

   Two matrices can be added only if they have the same order; addition is done entry-wise: $(A+B)_{ij} = a_{ij} + b_{ij}$.

4. State the condition for the product $AB$ to exist and the order of the result.

   If $A$ is $m \times n$ and $B$ is $n \times p$ (columns of $A$ = rows of $B$), then $AB$ exists and has order $m \times p$.

5. Is matrix multiplication commutative? State the key property.

   No. In general $AB \neq BA$; matrix multiplication is associative and distributive over addition but not commutative.

6. Define the transpose of a matrix and give $(AB)^{t}$.

   The transpose $A^{t}$ is formed by interchanging rows and columns. The reversal law holds: $(AB)^{t} = B^{t} A^{t}$.

7. Give the formula for the determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

   $\det A = |A| = ad - bc$.

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

   A square matrix is singular if $\det A = 0$ (no inverse exists) and non-singular if $\det A \neq 0$ (an inverse exists).

9. Give the formula for the inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

   $A^{-1} = \dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $ad - bc \neq 0$.

10. What are the minor and cofactor of an element $a_{ij}$ of a matrix?

    The minor $M_{ij}$ is the determinant obtained by deleting row $i$ and column $j$. The cofactor is $A_{ij} = (-1)^{i+j} M_{ij}$.

11. What is the adjoint of a matrix, and how is $A^{-1}$ expressed using it?

    The adjoint is the transpose of the matrix of cofactors. Then $A^{-1} = \dfrac{1}{\det A}\,\text{adj}(A)$, for $\det A \neq 0$.

12. Write a system of linear equations in matrix form.

    A system can be written as $AX = B$, where $A$ is the coefficient matrix, $X$ the column matrix of variables, and $B$ the column matrix of constants.

13. Describe the matrix inversion method for solving $AX = B$.

    If $A$ is non-singular ($\det A \neq 0$), multiply both sides by $A^{-1}$ to get the unique solution $X = A^{-1}B$.

14. State Cramer's rule for a system $AX = B$.

    For a non-singular system, each unknown is $x_i = \dfrac{\det A_i}{\det A}$, where $A_i$ is $A$ with its $i$-th column replaced by the constants column $B$.

15. Using Cramer's rule for two equations, write $x$ and $y$ for $\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$.

    $x = \dfrac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}, \quad y = \dfrac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$, with the common denominator $\neq 0$.

16. What does it mean if $\det A = 0$ when solving a square linear system?

    The system is not uniquely solvable; it either has no solution (inconsistent) or infinitely many solutions (dependent equations).

17. State the quadratic formula for $ax^{2} + bx + c = 0$ ($a \neq 0$).

    $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

18. What is the discriminant of a quadratic equation, and what does it determine?

    The discriminant is $\Delta = b^{2} - 4ac$. Its sign determines the nature of the roots of $ax^{2}+bx+c=0$.

19. Describe the nature of the roots based on the value of the discriminant $b^{2}-4ac$.

    If $\Delta > 0$: two distinct real roots ($\Delta$ a perfect square $\Rightarrow$ rational, else irrational). If $\Delta = 0$: real and equal (repeated) roots. If $\Delta < 0$: complex conjugate roots.

20. When are the roots of $ax^{2}+bx+c=0$ (rational coefficients) rational?

    The roots are rational when the discriminant $b^{2}-4ac$ is a perfect square (and $\geq 0$); otherwise they are irrational (or complex).

21. For roots $\alpha$ and $\beta$ of $ax^{2}+bx+c=0$, give the sum and product of the roots.

    Sum: $\alpha + \beta = -\dfrac{b}{a}$. Product: $\alpha \beta = \dfrac{c}{a}$.

22. How do you form a quadratic equation given its roots $\alpha$ and $\beta$?

    $x^{2} - (\alpha + \beta)x + \alpha\beta = 0$, i.e. $x^{2} - (\text{sum of roots})x + (\text{product of roots}) = 0$.

23. Express $\alpha^{2} + \beta^{2}$ in terms of the sum and product of the roots.

    $\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta$.

24. If one root of a real-coefficient quadratic is complex, what is the other root?

    Complex roots occur in conjugate pairs; if one root is $a+bi$, the other is its conjugate $a-bi$.

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