Correct Answer - Option 2 : Same
If A is any square matrix of order n, we can form the matrix [A – λI], where I is the nth order unit matrix. The determinant of this matrix equated to zero i.e. |A – λI| = 0 is called the characteristic equation of A.
The roots of the characteristic equation are called Eigen values or latent roots or characteristic roots of matrix A.
Properties of Eigen values:
(1) If λ is an eigen value of a matrix A, then λn will be an eigen value of a matrix An.
(2) If λ is an eigen value of a matrix A, then kλ will be an eigen value of a matrix kA where k is a scalar.
(3) Sum of eigen values is equal to trace of that matrix.
(4) The product of Eigen values of a matrix A is equal to the determinant of that matrix A.
(5) If λ is an Eigen value of matrix A, then λ2 will be an Eigen value of matrix A2.
(6) If λ1 is an Eigen value of matrix A, then (λ1 + 1) will be an Eigen value of matrix (A + I).
(7) Eigen values of a matrix and its transpose are same because transpose matrix will also have same characteristic equation.
