Show AB = A and BA = B Show That \(AB = A\) and \(BA = B\) Question: If \[ A=\begin{bmatrix} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{bmatrix}, \quad B=\begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & […]
Show AB = BA = O₃×₃ Show That \(AB = BA = O_{3\times3}\) Question: If \[ A=\begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix}, \quad B=\begin{bmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{bmatrix}
Show AB = BA = O Show That \(AB = BA = O_{3\times3}\) Question: If \[ A=\begin{bmatrix} 2 & -3 & -5 \\ -1 & 4 & 5 \\ 1 & -3 & -4 \end{bmatrix}, \quad B=\begin{bmatrix} -1 & 3 & 5 \\ 1 & -3 & -5 \\ -1 & 3 & 5 \end{bmatrix}
Find A² (Trigonometric Matrix) Find \(A^2\) Question: If \[ A=\begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{bmatrix} \] find \(A^2\). Solution: Step 1: Multiply \(A \cdot A\) \[ A^2 = \begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{bmatrix} \begin{bmatrix} \cos 2\theta & \sin 2\theta \\
If A=[[cos2θ, sin2θ], [-sin2θ, cos2θ]], find A^2 Read More »
Prove A² = O Prove \(A^2 = O\) Question: If \[ A=\begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix} \] show that: \[ A^2 = O \] Solution: Step 1: Compute \(A^2 = A \cdot A\) \[ A^2 = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix} \begin{bmatrix} ab & b^2 \\ -a^2
If A=[[ab, b^2], [-a^2, -ab]], show tht A^2 = O. Read More »
Find A² and A³ Find \(A^2\) and \(A^3\) Question: If \[ A=\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix} \] show that: \[ A^2=\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}, \quad A^3=\begin{bmatrix}1 & 3 \\ 0 & 1\end{bmatrix} \] Solution: Step 1: Compute \(A^2 = A \cdot A\) \[ A^2 = \begin{bmatrix}1 & 1 \\ 0
If A=[[1, 1], [0, 1]], show that A^2 = [[1, 2], [0, 1]] and A^3 = [[1, 3], [0, 1]] Read More »
Prove (A − 2I)(A − 3I) = O Prove \((A – 2I)(A – 3I) = O\) Question: If \[ A=\begin{bmatrix}4 & 2 \\ -1 & 1\end{bmatrix} \] prove that: \[ (A – 2I)(A – 3I) = O \] Solution: Step 1: Identity Matrix \[ I=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} \] Step 2: Compute
If A = [[4, 2], [-1, 1]], prove that (A – 2I) (A – 3I) = O. Read More »
Find 3A² − 2B + I Evaluate \(3A^2 – 2B + I\) Question: If \[ A=\begin{bmatrix}2 & -1 \\ 3 & 2\end{bmatrix}, \quad B=\begin{bmatrix}0 & 4 \\ -1 & 7\end{bmatrix} \] find: \[ 3A^2 – 2B + I \] Solution: Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix}2 & -1 \\ 3 & 2\end{bmatrix} \begin{bmatrix}2
If A = [[2, -1], [3, 2]] and B = [[0, 4], [-1, 7]], find 3A^2 – 2B + I. Read More »
Prove A² = B² = C² = I₂ Prove \(A^2 = B^2 = C^2 = I_2\) Question: Given \[ A=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, \quad B=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}, \quad C=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \] show that: \[ A^2 = B^2 = C^2 = I_2 \] Solution:
Evaluate A(B – C) Evaluate \(A(B – C)\) Question: Evaluate: \[ \begin{bmatrix} 1 & -1 \\ 0 & 2 \\ 2 & 3 \end{bmatrix} \left( \begin{bmatrix} 1 & 0 & 2 \\ 2 & 0 & 1 \end{bmatrix} – \begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 2 \end{bmatrix} \right) \] Solution: