Scalar Matrix Multiplication Property 📘 Question If \(S = [s_{ij}]\) is a scalar matrix such that \(s_{ii} = k\), and \(A\) is a square matrix of the same order, then: \[ AS = SA = \; ? \] (a) \(Ak\) (b) \(k + A\) (c) \(kA\) (d) \(kS\) ✏️ Step-by-Step Solution Step 1: Understand scalar […]
Find Relation for A² = I 📘 Question If \[ A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} \] and \(A^2 = I\), find the relation between \(\alpha, \beta, \gamma\). ✏️ Step-by-Step Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix} \begin{bmatrix} \alpha &
If A = [[α, β], [γ, -α]] is such that A^2 = I, then Read More »
Find a and b Using Matrix Identity 📘 Question If \[ A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix} \] and \[ (A + B)^2 = A^2 + B^2 \] Find the values of \(a\) and \(b\). ✏️ Step-by-Step Solution
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 =
If A = [[1, a], [0, 1]], then A^n (where n∈N) equals Read More »
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
Let A = [[a, 0, 0], [0, a, 0], [0, 0, a]], then A^n is equal to Read More »
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: