Basic Maths - Grand Test 1 -Q9

+4 votes

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}

asked Jun 21 in Basic Maths by gbeditor (21,290 points)
reshown Jun 23 by gbeditor

3 Answers

+3 votes
 
Best answer

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 gbeditor (21,290 points)
0 votes
D is correct one.
answered Jun 23 by tsnikhilsharmagate2018 (24,690 points)
0 votes

D is correct one.

answered Jun 23 by tsnikhilsharmagate2018 (24,690 points)
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