Educational

Find Matrix A Question Find matrix \(A\) such that \[ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} A = \begin{bmatrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{bmatrix} \] Solution Step 1: Assume \(A\) \[ A = \begin{bmatrix} a & b & c \\ d & e & f […]

Find Unknown Matrix Without Inverse Question Without using inverse, find \[ \begin{bmatrix} x & y \\ z & u \end{bmatrix} \] such that \[ \begin{bmatrix} 5 & -7 \\ -2 & 3 \end{bmatrix} \begin{bmatrix} x & y \\ z & u \end{bmatrix} = \begin{bmatrix} -16 & -6 \\ 7 & 2 \end{bmatrix} \] Solution Step

Prove A² − 7A + 10I₃ = O Question If \[ A = \begin{bmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix} \] show that \[ A^2 – 7A + 10I_3 = O. \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 11 &

Prove A² − 4A + 5I = O Question If \[ A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} \] prove that \[ A^2 – 4A + 5I = O. \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 9 &

Show A is Root of Polynomial Question If \[ A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \] show that \(A\) is a root of the polynomial \[ f(x)=x^3-6x^2+7x+2. \] Solution Step 1: Write \(f(A)\) \[ f(A)=A^3 – 6A^2 + 7A +

Find f(A) for Matrix Question If \[ f(x)=x^3+4x^2-x \] and \[ A = \begin{bmatrix} 0 & 1 & 2 \\ 2 & -3 & 0 \\ 1 & -1 & 0 \end{bmatrix} \] find \(f(A)\). Solution Step 1: Write Expression \[ f(A)=A^3 + 4A^2 – A \] Step 2: Compute \(A^2\) \[ A^2 = \begin{bmatrix}

Find f(A) for Matrix Question If \[ f(x)=x^2-2x \] and \[ A = \begin{bmatrix} 0 & 1 & 2 \\ 4 & 5 & 0 \\ 0 & 2 & 3 \end{bmatrix} \] find \(f(A)\). Solution Step 1: Write Expression \[ f(A)=A^2 – 2A \] Step 2: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 4 &

Compute A² − 4A + 3I₃ Question If \[ A = \begin{bmatrix} 1 & 2 & 0 \\ 3 & -4 & 5 \\ 0 & -1 & 3 \end{bmatrix} \] compute \[ A^2 – 4A + 3I_3. \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 7 & -6 & 10 \\ -9

Solve Matrix Equation Question Solve the matrix equation: \[ [2x\ \ 3] \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix} \begin{bmatrix} x \\ 8 \end{bmatrix} = 0 \] Solution Step 1: Single Matrix Equation \[ [2x\ \ 3] \begin{bmatrix} x + 16 \\ -3x \end{bmatrix} = 2x(x+16) + 3(-3x) = 0 \] Step 2:

Solve Matrix Equation Question Solve the matrix equation: \[ [x\ \ -5\ \ -1] \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = 0 \] Solution Step 1: Single Matrix Equation \[ [x\ \ -5\ \ -1]