SmartSchoolAI – Quiz Orientation

Before you begin the quiz, review this reference sheet that summarizes key concepts from the Mathematics-I syllabus (MTL-101) for B.Tech (Computer Science & Engineering). These concepts are foundational for engineering students and form the basis of many of the problems you’ll encounter.

Chapter 1 – Introduction to Matrices

A matrix is a rectangular array of elements arranged in rows and columns.
Denoted as \( A = [a_{ij}]_{m \times n} \)
Types: Square, Zero, Row, Column, Diagonal, Identity.

Chapter 2 – Inverse and Rank of a Matrix

The inverse \( A^{-1} \) exists if \( \det(A) \neq 0 \).
Rank = maximum number of linearly independent rows/columns.
Find inverse using adjoint: \( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \)

Chapter 3 – Rank-Nullity Theorem

For matrix \( A \in \mathbb{R}^{m \times n} \):
\( \text{Rank}(A) + \text{Nullity}(A) = n \)
Nullity is the dimension of the solution space of \( Ax = 0 \).

Chapter 4 – Symmetric, Skew-Symmetric & Orthogonal Matrices

Symmetric: \( A^T = A \)
Skew-Symmetric: \( A^T = -A \), diagonal entries are zero.
Orthogonal: \( A^T A = I \), thus \( A^{-1} = A^T \)

Chapter 5 – Hermitian & Skew-Hermitian Matrices

Hermitian: \( A = A^H \)
Skew-Hermitian: \( A = -A^H \)
Diagonal entries of Hermitian matrices are real; skew-Hermitian are imaginary or zero.

Chapter 6 – Unitary Matrix

A matrix \( A \) is unitary if \( A^H A = I \)
Generalization of orthogonal matrices to complex fields.

Chapter 7 – Determinants

Scalar value used for computing inverse, volume scaling, and singularity.
\( \det(AB) = \det(A) \cdot \det(B) \), \( \det(A^T) = \det(A) \)

Chapter 8 – System of Linear Equations

Linear system \( Ax = b \)
Solved using: Gauss elimination, Cramer’s Rule, Matrix Inversion.

Chapter 9 – Eigenvalues and Eigenvectors

\( Ax = \lambda x \), where \( \lambda \) is an eigenvalue, \( x \) is the eigenvector.
Characteristic equation: \( \det(A - \lambda I) = 0 \)

Chapter 10 – Diagonalization of Matrices

Convert \( A \) into \( D \) such that \( A = PDP^{-1} \)
Only diagonalizable if there are enough linearly independent eigenvectors.

Chapter 11 – Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic equation.
Used to compute powers or inverses efficiently.
Example: If \( A^2 - 5A + 6I = 0 \), then substitute \( A \) back into expression.

Chapter 16 – D'Alembert's Ratio Test

D’Alembert’s Ratio Test—also known simply as the Ratio Test.
It is a powerful tool for determining whether an infinite series converges or diverges.
It’s especially handy when dealing with series whose terms involve factorials, exponentials, or powers..
🧮 The Test Statement Given a series:∑𝑛=1∞𝑎𝑛

Best of luck! 🚀 Take your time to understand the questions. You can refer back to this page at any point during your quiz.

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