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
Matrix Power and Identity Question If \[ A = \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} \] find \[ (i)\ A^2 – 5A – 14I,\quad (ii)\ A^2 – 5A + 14I,\quad (iii)\ A^3 \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 29 & -25 \\ -20 & 24 \end{bmatrix} \] Step
Verify A² + A = A(A + I) Question If \[ A = \begin{bmatrix} 1 & 0 & -3 \\ 2 & 1 & 3 \\ 0 & 1 & 1 \end{bmatrix} \] verify that \[ A^2 + A = A(A + I) \] where \(I\) is the identity matrix. Solution Step 1: Compute \(A^2\)
Prove (A + B)² = A² + B² Question If \[ A = \begin{bmatrix} 0 & -x \\ x & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \] and \(x^2 = -1\), show that \[ (A + B)^2 = A^2 + B^2. \] Solution Step 1: Compute \(A^2\)
Find A¹⁶ for Matrix Question If \[ A = \begin{bmatrix} 0 & 0 \\ 4 & 0 \end{bmatrix} \] find \(A^{16}\). Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 0 & 0 \\ 4 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 4 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 &
If A = [[0, 0], [4, 0]], find A^16 Read More »
Find 2×2 Matrix A Question Find a \(2 \times 2\) matrix \(A\) such that \[ A \begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix} = 6I_2 \] Solution Step 1: Assume \(A\) \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] Step 2: Multiply \[ A \begin{bmatrix} 1 & -2
Find a 2×2 matrix A such that A[[1, -2], [1, 4]] = 6I2 Read More »
Find Matrix A Question Find matrix \(A\) such that \[ A \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \\ 11 & 10 & 9 \end{bmatrix} \] Solution Step 1: Assume \(A\) \[ A = \begin{bmatrix} a
Find Matrix A Question Find matrix \(A\) such that \[ \begin{bmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{bmatrix} A = \begin{bmatrix} -1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15 \end{bmatrix} \] Solution Step 1: Assume \(A\) \[ A = \begin{bmatrix} a
Find Matrix A Question Find \(A\) such that \[ [2\ \ 1\ \ 3] \begin{bmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} = A \] Solution Step 1: Single Matrix Equation \[ [2\ \ 1\ \ 3]