Educational

Find A^4 for Scalar Matrix Find A4 Given: \[ A = \begin{bmatrix} -3 & 0 \\ 0 & -3 \end{bmatrix} \] Step 1: Recognize the Matrix \[ A = -3I \] Step 2: Compute A4 \[ A^4 = (-3I)^4 = (-3)^4 I^4 \] \[ A^4 = 81I \] Final Answer: \[ A^4 = \begin{bmatrix} 81 […]

Find A^3 for -I Matrix Find A3 Given: \[ A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \] Step 1: Recognize the Matrix \[ A = -I \] Step 2: Compute A3 \[ A^3 = (-I)^3 = (-1)^3 I^3 \] \[ A^3

Find A^2 for -I Matrix Find A2 Given: \[ A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \] Step 1: Recognize the Matrix \[ A = -I \] Step 2: Compute A2 \[ A^2 = (-I)^2 = (-1)^2 I^2 \] \[ A^2

Find λ such that A^4 = λA Find λ such that A4 = λA Given: \[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \] Step 1: Find A2 \[ A^2 = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix}

Find k such that A^2 = kA Find k such that A2 = kA Given: \[ A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \] Step 1: Compute A2 \[ A^2 = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \] \[

Find x and y in Matrix Equation Find x and y from Matrix Equation Given: \[ \begin{bmatrix} 1 & 0 \\ y & 5 \end{bmatrix} + 2\begin{bmatrix} x & 0 \\ 1 & -2 \end{bmatrix} = I \] \[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \] Step 1: Multiply the

Find AA^T for Trigonometric Matrix Find AAT Given: \[ A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix} \] Step 2: Compute AAT \[ AA^T = \begin{bmatrix} \cos

Find x when A + A^T = I Find x such that A + AT = I Given: \[ A = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix}

Find A^2 for Complex Matrix Find A2 Given: \[ A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \] Step 1: Compute A2 = A × A \[ A^2 = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \] \[ A^2 = \begin{bmatrix}

Find A + A^T Find A + AT Given: \[ A = \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} 2 & 5 \\ 3 & 7 \end{bmatrix} \] Step 2: Compute A + AT \[ A + A^T = \begin{bmatrix} 2 & 3 \\