Try the fastest way to create flashcards
Question

# Find the eigenvalues, and if necessary the corresponding eigenvectors, of A and determine whether A is diagonalizable.$A=\left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 \end{array}\right]$

Solution

Verified
Step 1
1 of 3

$A=\left[ \begin{array}{llll}{0} & {0} & {0} & {0} \\ {1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1} \\ {1} & {1} & {1} & {1}\end{array}\right]$

The eigenvalues will be solutions of characteristic equation. Since, we know the characteristic equation form is det $(A-\lambda I)=0$ thus,

$det\left[ \begin{array}{llll}{-\lambda} & {0} & {0} & {0} \\ {1} & {-\lambda} & {1} & {0} \\ {0} & {1} & {-\lambda} & {1} \\ {1} & {1} & {1} & {1-\lambda}\end{array}\right]=0$

$-\lambda\left[ \begin{array}{lll}{-\lambda} & {1} & {0} \\ {1} & {-\lambda} & {1} \\ {1} & {1} & {1-\lambda} \end{array}\right]=0$

$\lambda^{4}-\lambda^{3}-2\lambda^{2}=0$

$\lambda=0,-1,2$

Eigenvalues are $0,-1$ and $2$ with multiplicity $2$, the eigenvector of corresponding eigenvalues for $\lambda =0$

$\left[ \begin{array}{llll}{0} & {0} & {0} & {0} \\ {1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1} \\ {1} & {1} & {1} & {1}\end{array}\right]\left[ \begin{array}{rrrr}{x} \\ {y} \\{z}\\{u}\end{array}\right]=\left[ \begin{array}{rrrr}{0} \\ {0}\\{0}\\{0} \end{array}\right]$

$x=-z$

$y=-u$

So, eigenvector will be

$v_{1}=\left\{t\left[ \begin{array}{rrrr}{1} \\ {0}\\{-1} \\{0}\end{array}\right]\hskip 0.5em t\epsilon R \hskip 0.5em s\left[ \begin{array}{rrrr}{0} \\ {1}\\{0} \\{-1}\end{array}\right]\hskip 0.5em s\epsilon R \right\}$



## Recommended textbook solutions

#### Linear Algebra and Its Applications

5th EditionISBN: 9780321982384 (3 more)David C. Lay, Judi J. McDonald, Steven R. Lay
2,070 solutions

#### Elementary Linear Algebra

11th EditionISBN: 9781118473504Howard Anton
2,932 solutions

#### Elementary Linear Algebra

12th EditionISBN: 9781119406778Anton Kaul, Howard Anton
3,078 solutions

#### Introduction to Linear Algebra with Applications

1st EditionISBN: 9781478627777 (3 more)Daniel Gagliardi, James DeFranza
1,494 solutions