PYQs Based Test - 02 : Eigenvalues and Eigenvectors

Consider the following matrix.

≤ft( {*20c 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 } )

The largest eigenvalue of the above matrix is ______

If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ³ – 4λ² + aλ + 30, a ∈ R, and one eigenvalue of M is 2, then the largest among the absolute value of the eigenvalues of M is __________.

The number of linearly independent eigen vectors of matrix A = ≤ft[ {*20c 2&1&0\\ 0&2&0\\ 0&0&3 } ] is ________

Consider the matrix A = ≤ft( {*20c 2&1&1\\ 2&3&4\\ - 1& - 1& - 2 } ) whose eigenvalues are 1, -1 and 3. Then Trace of (A³ – 3A²) is _________.

The eigenvectors of the matrix ≤ft[ {*20c 1&2\\ 0&2 } ] are written in the form ≤ft[ {*20c 1\\ a } ]and≤ft[ {*20c 1\\ b } ] . What is (a + b) ?