Ravi Kant Kumar

Find A² + B² Using Matrix Identities 📘 Question If matrices \(A\) and \(B\) satisfy: \[ AB = B \quad \text{and} \quad BA = A \] Find the value of: \[ A^2 + B^2 \] Options: (a) \(2AB\) (b) \(2BA\) (c) \(A + B\) (d) \(AB\) ✏️ Step-by-Step Solution Step 1: Find \(A^2\) From \(BA […]

Prove A² = A and B² = B 📘 Question If \(A\) and \(B\) are square matrices such that: \[ AB = A \quad \text{and} \quad BA = B \] Prove that: \[ A^2 = A \quad \text{and} \quad B^2 = B \] ✏️ Step-by-Step Solution Part 1: Prove \(A^2 = A\) Given: \[ AB

Find B² Using Matrix Identities 📘 Question If matrices \(A\) and \(B\) satisfy: \[ AB = A \quad \text{and} \quad BA = B \] Find \(B^2\). Options: (a) \(B\) (b) \(A\) (c) \(I\) (d) \(0\) ✏️ Step-by-Step Solution Step 1: Use given relation \[ BA = B \] Step 2: Multiply both sides by \(B\)

Find A² Using Single Matrix Multiplication 📘 Question If \[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \end{bmatrix} \] find \(A^2\). ✏️ Solution (Single-Step Matrix Multiplication) \[ A^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\

Find A^(4n) for A = iI 📘 Question If \[ A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = iI \] Find \(A^{4n}\), where \(n \in \mathbb{N}\). ✏️ Step-by-Step Solution Step 1: Express matrix \[ A = iI \] — Step 2: Use power rule \[ A^{4n} = (iI)^{4n} = i^{4n} \cdot

Find Order of Matrix AB 📘 Question Let \(A\) and \(B\) be matrices of orders \(3 \times 2\) and \(2 \times 4\) respectively. Find the order of matrix \(AB\). ✏️ Step-by-Step Solution Step 1: Check multiplication condition Matrix multiplication is possible if the number of columns of \(A\) equals the number of rows of \(B\).

Find Symmetric Matrix P 📘 Question If \[ A = \begin{bmatrix} 3 & 5 \\ 7 & 9 \end{bmatrix} \] is written as \(A = P + Q\), where \(P\) is symmetric and \(Q\) is skew-symmetric, find matrix \(P\). ✏️ Step-by-Step Solution Step 1: Use decomposition formula For any matrix: \[ P = \frac{A +

Find Order of Matrix A 📘 Question If \[ [2 \;\; 1 \;\; 3] \begin{bmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} = A \] Find the order of matrix \(A\). ✏️ Step-by-Step Solution Step 1: Identify

Number of 2×2 Matrices with Entries 1, 2, 3 📘 Question Write the number of all possible matrices of order \(2 \times 2\) with each entry being 1, 2, or 3. ✏️ Step-by-Step Solution A \(2 \times 2\) matrix has 4 entries: \[ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \] Each entry

Find a and b for Symmetric Matrix 📘 Question Matrix \[ A = \begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix} \] is symmetric. Find the values of \(a\) and \(b\). ✏️ Step-by-Step Solution Step 1: Use symmetry condition For a symmetric matrix: \[