Ravi Kant Kumar

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 \(|A| = 0\)

Find Order of Matrix B 📘 Question If \(A\) is a \(3 \times 4\) matrix and \(B\) is such that both: \[ A^T B \quad \text{and} \quad BA^T \] are defined, find the order of matrix \(B\). (a) \(3 \times 4\) (b) \(3 \times 3\) (c) \(4 \times 4\) (d) \(4 \times 3\) ✏️ Step-by-Step

Find x and y for Symmetric Matrix 📘 Question If \[ A = \begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix} \] and \(A = A^T\), find \(x\) and \(y\). ✏️ Step-by-Step Solution Step 1: Use symmetric condition \[ A = A^T \] So corresponding elements must be equal: \(x = y\) Step 2:

ABA Symmetric Matrix Property 📘 Question If \(A\) and \(B\) are symmetric matrices, then \(ABA\) is: (a) symmetric matrix (b) skew-symmetric matrix (c) diagonal matrix (d) scalar matrix ✏️ Step-by-Step Solution Step 1: Use symmetry condition \[ A^T = A,\quad B^T = B \] Step 2: Take transpose of \(ABA\) \[ (ABA)^T = A^T B^T

AAᵀ is Symmetric Matrix 📘 Question If \(A\) is a square matrix, then \(AA’\) (or \(A A^T\)) is: (a) skew-symmetric matrix (b) symmetric matrix (c) diagonal matrix (d) none of these ✏️ Step-by-Step Solution Step 1: Use transpose property \[ (AA^T)^T = (A^T)^T A^T \] Step 2: Simplify \[ = A A^T \] Step 3:

Identify Skew-Symmetric Matrix 📘 Question The matrix \[ \begin{bmatrix} 0 & 5 & -7 \\ -5 & 0 & 11 \\ 7 & -11 & 0 \end{bmatrix} \] is: (a) a skew-symmetric matrix (b) a symmetric matrix (c) a diagonal matrix (d) an upper triangular matrix ✏️ Step-by-Step Solution Step 1: Recall definition A matrix

Symmetric and Skew-Symmetric Matrix MCQ 📘 Question If a matrix \(A\) is both symmetric and skew-symmetric, then: (a) \(A\) is a diagonal matrix (b) \(A\) is a zero matrix (c) \(A\) is a scalar matrix (d) \(A\) is a square matrix ✏️ Step-by-Step Solution Step 1: Use definitions Symmetric matrix: \(A^T = A\) Skew-symmetric matrix:

Find (I + A)³ – 7A 📘 Question If a square matrix \(A\) satisfies: \[ A^2 = A \] Find the value of: \[ (I + A)^3 – 7A \] (a) \(A\) (b) \(I – A\) (c) \(I\) (d) \(3A\) ✏️ Step-by-Step Solution Step 1: Expand \((I + A)^3\) \[ (I + A)^3 = I