Question

If Ais a $\mathbf{2}\mathbf{\times }\mathbf{2}$matrix with eigenvalues 3 and 4 and if localid="1668109698541" $\overrightarrow{\mathbf{u}}$ is a unit eigenvector of A, then the length of vector Alocalid="1668109419151" $\overrightarrow{\mathbf{u}}$cannot exceed 4.

Short answer

The given statement is True because the given matrices are not exceeded 4.

Step-by-step solution

Definition of Eigenvalue

The dedication of a system's eigenvalues and eigenvectors is vital in physics and engineering, in which it's far corresponding to matrix diagonalization and takes place in programs as various as balance analysis, rotating frame physics, and tiny oscillations of vibrating systems, to say a few.

Determine whether the given statement is true or false

Each eigenvalue has an eigenvector that corresponds to it (or, in general, a corresponding proper eigenvector and a corresponding left eigenvector; there may be no analogous difference among left and proper for eigenvalues).

Let be a unit eigenvector of A.

If its corresponding eigenvalue is 3.

Then,

$||Au||=||3u||=3\le 4$.

If the corresponding eigenvalue is though,

Then,

$||Au||=||4u||=4\le 4$.

Therefore, the given statement is True.