Viva Questions:

Can you multiply a $2 \times 3$ matrix by a $3 \times 2$ matrix? What is the order of the resulting matrix?
What is an identity matrix?
If $A$ is a square matrix, can it be both symmetric and skew-symmetric?
What is the significance of $A A^{-1} = I$?
What is the difference between a singular and a non-singular matrix?

4. Determinants

Determinant of a $2 \times 2$ matrix: $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \implies |A| = ad - bc$.
Determinant of a $3 \times 3$ matrix: Using cofactor expansion along any row or column.
Properties of Determinants:
- $|A^T| = |A|$
- If two rows/columns are identical or proportional, $|A|=0$.
- If any row/column consists of all zeros, $|A|=0$.
- If rows/columns are interchanged, determinant changes sign.
- If a row/column is multiplied by $k$, determinant is multiplied by $k$.
- If $A$ is $n \times n$, then $|kA| = k^n|A|$.
- $|AB| = |A||B|$.
- Operation $R_i \to R_i + k R_j$ (or $C_i \to C_i + k C_j$) does not change the determinant value.
Minors and Cofactors:
- Minor $M_{ij}$: determinant of submatrix by deleting $i$-th row and $j$-th column.
- Cofactor $A_{ij} = (-1)^{i+j} M_{ij}$.
Adjoint of a Matrix: $adj(A) = (A_{ij})^T$.
Inverse of a Matrix: $A^{-1} = \frac{1}{|A|} adj(A)$. Exists if $|A| \neq 0$.
Theorem: $A \cdot (adj A) = (adj A) \cdot A = |A|I$.
Solving System of Linear Equations (Matrix Method $AX=B$):
- If $|A| \neq 0$: Unique solution $X = A^{-1}B$. (Consistent)
- If $|A| = 0$:
  - If $(adj A)B \neq O$: No solution. (Inconsistent)
  - If $(adj A)B = O$: Infinitely many solutions. (Consistent, dependent)
Area of a Triangle: $\frac{1}{2} | \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} |$. Collinear points if determinant is 0.
Viva Questions:
- What is the condition for a matrix to be invertible?
- If two rows of a determinant are identical, what is its value? Why?
- Explain the difference between a minor and a cofactor.
- How does a determinant help in checking the consistency of a system of linear equations?
- What happens to the determinant if you multiply an entire matrix by a scalar $k$?

5. Continuity and Differentiability

Continuity: A function $f(x)$ is continuous at $x=a$ if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$.
Differentiability: A function $f(x)$ is differentiable at $x=a$ if $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists. This limit is $f'(a)$.
Relation: Differentiability $\implies$ Continuity, but not vice-versa (e.g., $|x|$ at $x=0$).
Differentiation Formulas:
- $\frac{d}{dx}(c) = 0$
- $\frac{d}{dx}(x^n) = nx^{n-1}$
- $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$
- $\frac{d}{dx}(\tan x) = \sec^2 x$, $\frac{d}{dx}(\cot x) = -\csc^2 x$
- $\frac{d}{dx}(\sec x) = \sec x \tan x$, $\frac{d}{dx}(\csc x) = -\csc x \cot x$
- $\frac{d}{dx}(e^x) = e^x$, $\frac{d}{dx}(a^x) = a^x \log a$
- $\frac{d}{dx}(\log_e x) = \frac{1}{x}$, $\frac{d}{dx}(\lo