Educational

Trace of Scalar Matrix 📘 Question If \(A = [a_{ij}]\) is a scalar matrix of order \(n \times n\) such that \(a_{ii} = k\), then find \(\text{Trace}(A)\). (a) \(nk\) (b) \(n + k\) (c) \(\frac{n}{k}\) (d) none of these ✏️ Step-by-Step Solution Step 1: Understand scalar matrix All diagonal elements are equal: \[ a_{11} = […]

Find Trace of Matrix 📘 Question Find the trace of the matrix: \[ A = \begin{bmatrix} 1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9 \end{bmatrix} \] ✏️ Step-by-Step Solution Step 1: Definition Trace of a matrix is the sum of its diagonal elements: \[ \text{Trace}(A) =

Express B in Terms of I and J 📘 Question Given: \[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad J = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \] and \[ B = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \] Express \(B\) in terms of

Find k, a, b Using kA 📘 Question If \[ A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix} \] and \[ kA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix} \] Find \(k, a, b\). ✏️ Step-by-Step Solution Step 1: Multiply matrix by scalar \[ kA = k \begin{bmatrix} 0

Find x and y Using Matrix Equality 📘 Question Find \(x\) and \(y\) such that: \[ \begin{bmatrix} 3x + 7 & 5 \\ y + 1 & 2 – 3x \end{bmatrix} = \begin{bmatrix} 0 & y – 2 \\ 8 & 4 \end{bmatrix} \] ✏️ Step-by-Step Solution Step 1: Compare corresponding elements \(3x + 7

Number of 3×3 Matrices 📘 Question The number of all possible matrices of order \(3 \times 3\) with each entry 0 or 1 is: (a) 27 (b) 18 (c) 81 (d) 512 ✏️ Step-by-Step Solution Step 1: Total entries A \(3 \times 3\) matrix has: \[ 3 \times 3 = 9 \text{ entries} \] Step

Identify Scalar Matrix 📘 Question Out of the following matrices, choose the matrix which is a scalar matrix: (a) \(\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\) (b) \(\begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\) (c) \(\begin{bmatrix}0 & 0 \\ 0 & 0 \\ 0 & 0\end{bmatrix}\) (d) \(\begin{bmatrix}0 \\ 0 \\

Find Symmetric Matrix from A 📘 Question If \[ A = \begin{bmatrix} 2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2 \end{bmatrix} \] is expressed as the sum of a symmetric and skew-symmetric matrix, find the symmetric matrix. ✏️ Step-by-Step Solution Step 1: Use formula \[ S

Find θ for Aᵀ + A = I₂ 📘 Question If \[ A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \] then \(A^T + A = I_2\), if (a) \(\theta = n\pi\) (b) \(\theta = \frac{(2n+1)\pi}{2}\) (c) \(\theta = 2n\pi + \frac{\pi}{3}\) (d) none of these ✏️ Step-by-Step Solution Step 1: Find

Matrix aij = i² – j² Type 📘 Question If \(A = [a_{ij}]\) is a square matrix of even order such that: \[ a_{ij} = i^2 – j^2 \] Then: (a) \(A\) is skew-symmetric and \(|A| = 0\) (b) \(A\) is symmetric and \(|A|\) is a square (c) \(A\) is symmetric and \(|A| = 0\)