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 […]
If A = diag(a, b, c), show that A^n = diag(a^n, b^n, c^n) for all positive integer n. Read More »
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) \[
Prove Aⁿ for Matrix Question If \[ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \] prove that \[ A^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \] for all positive integers \(n\). Solution (Mathematical Induction) Step 1: Base Case (n = 1) \[ A^1 = \begin{bmatrix} 1 &
If A = [[1, 1], [0, 1]], prove that A^n = [[1, n], [0, 1]] for all positive integers n. Read More »
Find A² − 5A + 4I and Matrix X Question If \[ A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix} \] find \[ A^2 – 5A + 4I \] and hence find matrix \(X\) such that \[ A^2 – 5A + 4I
Prove PQ = QP for Diagonal Matrices Question If \[ P = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}, \quad Q = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \] prove
Prove P(x)P(y) = P(x+y) Question If \[ P(x)= \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix} \] show that \[ P(x)P(y)=P(x+y)=P(y)P(x). \] Solution Step 1: Compute \(P(x)P(y)\) \[ P(x)P(y)= \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix} \begin{bmatrix} \cos y & \sin y \\ -\sin