Question

For the matrix A, find (if possible) a nonsingular matrix P such that p^-1 AP is diagonal. (If not possible, enter IMPOSSIBLE.) 

\(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ -5 & -3 & 4 \\ -4 & 0 & -3\end{array}\right]\)

Verify that p^-1AP is a diagonal matrix with the eigenvalues on the main diagonal. 

Answer 1

Suola we have matrix 0 A = 0 4 -5 -3 sreb I find the eigenvalues of A. LA-d Il=0 I-X 0 0 -5 -3-1 4 =0 -4 6 -3-1 Expand along row I we get (1-1) (-34) (-3-2)=0 (2-1) (d+3)²=0 The eigenamees are with algebraic multiplicity 2 dezt with algesicic mutiplicity I d = -3 0 0 0 0 Step Ind the eigenrector of A. ford, = -3 11-1-3) -3-(-3) -3-(-3) -4 solve the matrix equation we get 4 -5 10 2 4 0 0 0 4 0 -ų 0 0 O 0

0 0 게 o 0 0 ~ o 0 0 0 o we get &z=0 21 b 0 0 of Mi? - 0 0 0 Hures geometria multiplicity of tarziu (number of eigenvectot). a alginraic mult. & geometuc mult. of dy=-3 g 1 = -3 So, number of eigenrectors of A mill not equal to the Wumns of A: Therefore, tre mateix Aů not diagonalizance exist. р does not

Answer 2

It's wrong