# Basic Maths - Grand Test 1 -Q9

Let -1,3 be the eigenvalues and  $\begin{bmatrix} 1 \\-1 \end{bmatrix}$$\begin{bmatrix} 1 \\1 \end{bmatrix}$ be the corresponding eigenvectors? of the matrix A, then which one of the following choices is TRUE ?

$(A).A =\begin{bmatrix} 1 & 1 \\-1& 1\end{bmatrix}\begin{bmatrix} -1 & 0 \\0& 3\end{bmatrix} \begin{bmatrix} 1 & -1 \\1& 1\end{bmatrix}$                $(B).A = \begin{bmatrix} 1 & 1 \\-1& 1\end{bmatrix}\begin{bmatrix} 3 & 0 \\0& -1\end{bmatrix} \begin{bmatrix} 1 & -1 \\1& 1\end{bmatrix}$

$(C).A = \begin{bmatrix} 1 & 1 \\-1& 1\end{bmatrix}\begin{bmatrix} 3 & 0 \\0& -1\end{bmatrix} \begin{bmatrix} \frac{1}{2} & -\frac{1}{2} \\\frac{1}{2}& \frac{1}{2}\end{bmatrix}$               $(D).A = \begin{bmatrix} 1 & 1 \\-1& 1\end{bmatrix}\begin{bmatrix} -1 & 0 \\0& 3\end{bmatrix} \begin{bmatrix} \frac{1}{2} & -\frac{1}{2} \\\frac{1}{2}& \frac{1}{2}\end{bmatrix}$

reshown Jun 23

+1 vote

We have ;$A = P DP^{-1}$ where P is a matrix of order 2 constructed by the linearly dependent eigenvectors of matrix A as column vectors and D is a diagonal matrix such that the diagonal elements as the eigenvalues of matrix A

so

$A = \begin{bmatrix} 1 & 1\\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & 3 \end{bmatrix} \begin{bmatrix} \frac{1}{2} & -\frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} \end{bmatrix}$

answered Jun 24 by (4,380 points)