Educational

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

Solve Matrix Equation Question Solve the matrix equation: \[ [x\ \ 1] \begin{bmatrix} 1 & 0 \\ -2 & -3 \end{bmatrix} \begin{bmatrix} x \\ 5 \end{bmatrix} = 0 \] Solution Step 1: Single Matrix Equation \[ [x\ \ 1] \begin{bmatrix} x \\ -2x – 15 \end{bmatrix} = x(x) + 1(-2x – 15) = 0 \]

Find x for Identity Matrix Question Find the value of \(x\) for which \[ \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix} \begin{bmatrix} -x & 14 & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix} = I \]

Find λ and μ in Matrix Equation Question If \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}, \quad I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \] find \( \lambda, \mu \) such that \[ A^2 = \lambda A + \mu I. \] Solution Step 1: Compute \(A^2\)

Show that f(A) = O Question If \[ A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \] and \( f(x)=x^2-2x-3 \), show that \( f(A)=O \). Solution Step 1: Write \(f(A)\) \[ f(A) = A^2 – 2A – 3I \] Step 2: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 1 & 2 \\

Find k in Matrix Equation Question If \[ A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} \] find \(k\) such that \[ A^2 – 8A + kI = O. \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} \begin{bmatrix} 1 & 0 \\

Find k in Matrix Equation Question If \[ A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} \] find \(k\) such that \[ A^2 = kA – 2I_2. \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix} \begin{bmatrix} 3 & -2 \\ 4 &

Find A⁴ using Matrix Identity Question If \[ A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \] show that \[ A^2 – 5A + 7I = O \] and hence find \(A^4\). Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 3 &

Find A² − 5A + 14I Question If \[ A = \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} \] find \[ A^2 – 5A + 14I. \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix}

Prove A² − 12A − I = O Question Show that the matrix \[ A = \begin{bmatrix} 5 & 3 \\ 12 & 7 \end{bmatrix} \] is a root of the equation \[ A^2 – 12A – I = O. \] Solution Step 1: Compute \(A^2\) \[ A^2 = \begin{bmatrix} 5 & 3 \\ 12