Educational

Example AB = O but A ≠ O, B ≠ O Question Give an example of matrices \(A\) and \(B\) such that \[ AB = O \quad \text{but} \quad A \ne O,\ B \ne O. \] Solution Step 1: Take Matrices \[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \quad B […]

Example AB ≠ BA Question Give an example of matrices \(A\) and \(B\) such that \[ AB \ne BA. \] Solution Step 1: Take Matrices \[ A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \] Step 2: Compute \(AB\) \[

Matrix Dimension Problem Question Matrix \(X\) has \((a+b)\) rows and \((a+2)\) columns. Matrix \(Y\) has \((b+1)\) rows and \((a+3)\) columns. Both \(XY\) and \(YX\) exist. Find \(a\) and \(b\). Also determine whether \(XY\) and \(YX\) are of same type and equal. Solution Step 1: Condition for \(XY\) to exist \[ \text{Columns of } X =

Transpose Power Identity Question If \(A\) is a square matrix, prove using mathematical induction that \[ (A^T)^n = (A^n)^T \quad \forall n \in \mathbb{N}. \] Solution (Mathematical Induction) Step 1: Base Case (n = 1) \[ (A^T)^1 = A^T \quad \text{and} \quad (A^1)^T = A^T \] ✔ True for \(n=1\) Step 2: Assume for \(n

Power of Diagonal Matrix Question If \[ A = \text{diag}(a, b, c) = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \] show that \[ A^n = \text{diag}(a^n, b^n, c^n) \] for all positive integers \(n\). Solution (Mathematical Induction) Step 1: Base Case (n

Matrix Induction with Nilpotent Matrix Question If \(A = B + C\), where \(BC = CB\) and \(C^2 = O\), prove that for every \(n \in \mathbb{N}\), \[ A^{n+1} = B^n (B + (n+1)C). \] Solution (Mathematical Induction) Step 1: Base Case (n = 1) \[ A^2 = (B + C)^2 = B^2 + BC

Prove Aⁿ for 3×3 Matrix Question If \[ A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} 1 & n & \frac{n(n+1)}{2} \\ 0 & 1 & n \\ 0 & 0 & 1 \end{bmatrix} \]

Prove Aⁿ Trigonometric Matrix Form Question If \[ A = \begin{bmatrix} \cos \alpha + \sin \alpha & \sqrt{2}\sin \alpha \\ -\sqrt{2}\sin \alpha & \cos \alpha – \sin \alpha \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} \cos n\alpha + \sin n\alpha & \sqrt{2}\sin n\alpha \\ -\sqrt{2}\sin n\alpha & \cos n\alpha – \sin n\alpha \end{bmatrix} \quad

Prove Aⁿ Trigonometric Matrix Question If \[ A = \begin{bmatrix} \cos \theta & i\sin \theta \\ i\sin \theta & \cos \theta \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} \cos n\theta & i\sin n\theta \\ i\sin n\theta & \cos n\theta \end{bmatrix} \quad \forall n \in \mathbb{N}. \] Solution (Mathematical Induction) Step 1: Base Case (n

Prove Aⁿ for General Matrix Question If \[ A = \begin{bmatrix} a & b \\ 0 & 1 \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} a^n & b\frac{a^n – 1}{a – 1} \\ 0 & 1 \end{bmatrix} \] for every positive integer \(n\). Solution (Mathematical Induction) Step 1: Base Case (n = 1) \[