(Solved): Linear algebra and differential equations MATRICS   1. Find the values of \( x, y, z \) ant ...

Linear algebra and differential equations

MATRICS

 

1. Find the values of \( x, y, z \) ant \( t \) witch satisfy the matrix equation \[ \left(\begin{array}{cc} x+1 & y-1 \\ 2 z-1 & t+3 \end{array}\right)=\left(\begin{array}{cc} 1 & 0 \\ 3 & -1 \end{array}\right) \] 2. Let \( A \) be a matrix given by \[ A=\left(\begin{array}{cc} -4 & 3 \\ 4-\lambda & 1 \end{array}\right) \] Find the value of \( \lambda \) such that \( \operatorname{det}(A)=0 \). 3. Find \( \alpha \) and \( \beta \) such that \[ \alpha\left(\begin{array}{ll} 1 & 2 \\ 4 & 1 \end{array}\right)+\beta\left(\begin{array}{ll} 2 & 1 \\ 3 & 1 \end{array}\right)=\left(\begin{array}{cc} -1 & 4 \\ 6 & 1 \end{array}\right) \] Question \( n^{\circ} 2 \) : Consider the matrix \( B \) given by \[ B=\left(\begin{array}{ccc} 1 & 1 & -1 \\ 1 & 0 & 3 \\ 2 & 3 & 1 \end{array}\right) \] 1. Calculate the minors and cofactors of \( B \). 2. Determine the determinant of \( B \). 3. Find the inverse of \( B \). Question \( n^{\circ} 3 \) : Let \( A \) be a matrix defined by \[ A=\left(\begin{array}{lll} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array}\right) \] 1. Prove that \( A^{2}=5 A-4 I_{3} \). 2. Deduce that \( A \) is invertible and then determine \( A^{-1} \).