Educational

Find x + y Using AB = I 📘 Question If \[ A = \begin{bmatrix} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} […]

Find Aⁿ for Upper Triangular Matrix 📘 Question If \[ A = \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix} \] Find \(A^n\), where \(n \in \mathbb{N}\). ✏️ Step-by-Step Solution Step 1: Observe pattern Compute first few powers: \[ A^2 = \begin{bmatrix} 1 & 2a \\ 0 & 1 \end{bmatrix} \] \[ A^3 =

Find AB for Scalar Matrix 📘 Question If \[ A = \begin{bmatrix} n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n \end{bmatrix} ,\quad B = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \] Find \(AB\).

AB = O and A Non-Singular 📘 Question If \(A\) and \(B\) are square matrices of order 3 such that: \[ AB = O \] and \(A\) is non-singular, then \(B\) is: (a) null matrix (b) singular matrix (c) unit matrix (d) non-singular matrix ✏️ Step-by-Step Solution Step 1: Use inverse of non-singular matrix Since

Find Aⁿ for Diagonal Matrix 📘 Question Let \[ A = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix} \] Find \(A^n\). ✏️ Step-by-Step Solution Step 1: Recognize the matrix This is a scalar matrix: \[ A = aI \] Step 2: Use power

AB = O Matrix Concept 📘 Question If \[ AB = O \] then which of the following is correct? (a) It is not necessary that either \(A = O\) or \(B = O\) (b) \(A = O\) or \(B = O\) (c) \(A = O\) and \(B = O\) (d) All the above statements

Find Least k for Rotation Matrix 📘 Question If \[ \left[ \begin{array}{cc} \cos\frac{2\pi}{7} & -\sin\frac{2\pi}{7} \\ \sin\frac{2\pi}{7} & \cos\frac{2\pi}{7} \end{array} \right]^k = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \] Find the least positive integer value of \(k\). ✏️ Step-by-Step Solution Step 1: Recognize the matrix This is a rotation matrix with angle:

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 \(B\)