Ravi Kant Kumar

Number System True or False 📘 Question Are the following statements true or false? Give reasons: (i) Every whole number is a natural number. (ii) Every integer is a rational number. (iii) Every rational number is an integer. (iv) Every natural number is a whole number. (v) Every integer is a whole number. (vi) Every […]

Identify Skew-Symmetric Matrix 📘 Question The matrix \[ A = \begin{bmatrix} 0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0 \end{bmatrix} \] is a: (a) diagonal matrix (b) symmetric matrix (c) skew-symmetric matrix (d) scalar matrix ✏️ Step-by-Step Solution Step 1: Check definition A matrix is skew-symmetric

Check AB and BA Defined 📘 Question If \[ A = \begin{bmatrix} 2 & -1 & 3 \\ -4 & 5 & 1 \end{bmatrix} \quad (2 \times 3) \] \[ B = \begin{bmatrix} 2 & 3 \\ 4 & -2 \\ 1 & 5 \end{bmatrix} \quad (3 \times 2) \] Then: (a) only AB is

(A+B)(A-B) Matrix Identity 📘 Question If \(A\) and \(B\) are square matrices of the same order, find: \[ (A + B)(A – B) \] (a) \(A^2 – B^2\) (b) \(A^2 – BA – AB – B^2\) (c) \(A^2 – B^2 + BA – AB\) (d) \(A^2 – BA + B^2 + AB\) ✏️ Step-by-Step Solution

Find A – B (Inverse Trig Matrix) 📘 Question If \[ A = \frac{1}{\pi} \begin{bmatrix} \sin^{-1}(\pi x) & \tan^{-1}(1) \\ \sin^{-1}\left(\frac{x}{\pi}\right) & \cot^{-1}(\pi x) \end{bmatrix} \] \[ B = \frac{1}{\pi} \begin{bmatrix} -\cot^{-1}(\pi x) & \tan^{-1}\left(\frac{x}{\pi}\right) \\ \sin^{-1}\left(\frac{x}{\pi}\right) & -\tan^{-1}(\pi x) \end{bmatrix} \] Find \(A – B\). ✏️ Step-by-Step Solution Step 1: Subtract matrices \[ A

Find A² from aij Rule 📘 Question If \(A = [a_{ij}]_{2 \times 2}\), where: \[ a_{ij} = \begin{cases} 1, & i \ne j \\ 0, & i = j \end{cases} \] Find \(A^2\). ✏️ Step-by-Step Solution Step 1: Construct matrix For \(2 \times 2\): \[ A = \begin{bmatrix} 0 & 1 \\ 1 & 0

ABᵀ – BᵀA is Skew-Symmetric 📘 Question If \(A\) and \(B\) are matrices of suitable order, then: \[ AB^T – B^T A \] is a: (a) skew-symmetric matrix (b) null matrix (c) unit matrix (d) symmetric matrix ✏️ Step-by-Step Solution Step 1: Take transpose \[ (AB^T – B^T A)^T = (AB^T)^T – (B^T A)^T \]

Find Order of Matrix B 📘 Question If \(A\) is of order \(m \times n\) and \(B\) is such that: \[ AB^T \quad \text{and} \quad B^T A \] are both defined, find the order of matrix \(B\). (a) \(m \times n\) (b) \(n \times n\) (c) \(n \times m\) (d) \(m \times n\) ✏️ Step-by-Step

Find Order of 5A – 2B 📘 Question If \(A\) and \(B\) are matrices of order \(3 \times m\) and \(3 \times n\) respectively and \(m = n\), find the order of: \[ 5A – 2B \] (a) \(m \times 3\) (b) \(3 \times 3\) (c) \(m \times n\) (d) \(3 \times n\) ✏️ Step-by-Step

Matrix Identity Expression 📘 Question If a square matrix \(A\) satisfies: \[ A^2 = I \] Find: \[ (A – I)^3 + (A + I)^3 – 7A \] (a) \(A\) (b) \(I – A\) (c) \(I + A\) (d) \(3A\) ✏️ Step-by-Step Solution Step 1: Use identity \[ (x-y)^3 + (x+y)^3 = 2x^3 + 6xy^2