91影视

For any n x n matrix A and a vector $\stackrel{鈫�}{u}鈭�{\mathrm{鈩�}}^n,$if $A^m\stackrel{鈫�}{u}=0$but $A^{m-1}\stackrel{鈫�}{u}鈮�0$we have 鈥榤鈥� linearly independent vectors $\stackrel{鈫�}{u},A\stackrel{鈫�}{u},A^2\stackrel{鈫�}{u},...,A^{m-1}\stackrel{鈫�}{u}$.

Here we have a 2 x 2 matrix A such that$A^3=0$ and$A^2鈮�0$ then for a vector$\stackrel{鈫�}{u}鈭�{\mathrm{鈩�}}^2,$ we have three linearly independent vectors$\stackrel{鈫�}{u},鈥�A\stackrel{鈫�}{u},鈥�A^2\stackrel{鈫�}{u}.$

Since matrix A is of order 2 x 2 then the rank of the matrix A can be at most two.

$鈬�$There can be at most two linearly independent vectors.

$鈬�$There doesn鈥檛 exist any 2 脳 2 matrix A such that $A^2鈮�0$and$A^3=0$ .

For any 2 x 2 matrix A, there can be at most two linearly independent vectors.

If A is a 2 x 2 matrix such that$A^3=0$ and$A^2鈮�0$ then there exist three linearly independent vectors which are not possible for a matrix A of order 2 x 2.

Hence, there doesn鈥檛 exist any 2 脳 2 matrices A such that$A^2鈮�0$ and$A^3=0$ .