Question

True or False? The matrix A = {{1, 0, 0, 0, 0},{2, -2, 0, 0, 0},{-1, 0, 3, 0, 0}{6, -9, 4, 2, 0},{7, 3, -2, 8, 5}} is diagonalizable.

Answer 1

$A=\begin{pmatrix}{\color{Red} 1}&0&0&0&0\\ 2&{\color{Red} -2}&0&0&0\\ -1&0&{\color{Red} 3}&0&0\\ 6&-9&4&{\color{Red} 2}&0\\ 7&3&-2&8&{\color{Red} 5}\end{pmatrix}$

as we can see here the matrix is in the triangular form

so eigenvalues are diagonal entries of the matrix A

so here eigenvalues are

${\color{Red} \:\lambda _1=1,\:\:\:\lambda _2=-2,\:\:\:\lambda _3=3,\:\:\:\lambda _4=2,\:\:\:\lambda _5=5}$

for one eigenvalue, we have at least one eigenvector so obviously there are 5 eigenvectors

to diagonalize the matrix, the number of eigenvectors should be equal to the columns of the matrix A

here we have 5 eigenvectors and the number of columns are also 5

so the matrix is diagonalizable

true