Question

Question: Show that if \({A^{\bf{2}}}\) is the zero matrix, then the only eigenvalue of A is 0.

Short answer

If the matrix \({A^2}\) is zero, then each eigenvalue of A is zero.

Step-by-step solution

Write the given information

For matrix A, the matrix \({A^2}\) is 0.

Check for the eigenvalue of \({A^{ - {\bf{1}}}}\)

As \({A^2}\) is a zero matrix, so the equation \(A{\bf{x}} = \lambda {\bf{x}}\) and \(x \ne 0\) can be represented as;

\(\begin{array}{c}{A^2}{\bf{x}} = A\left( {A{\bf{x}}} \right)\\ = A\left( {\lambda {\bf{x}}} \right)\\ = \lambda A{\bf{x}}\\ = {\lambda ^2}{\bf{x}}\end{array}\)

As x is nonzero, so \(\lambda \) must be zero.

Therefore, the eigenvalue of A is zero.