Correct Answer - Option 2 : 8, 12, 6 and 49, 121, 25
Concept:
Upper triangular matrix
When all the elements below the diagonal in the matrix are zero
- From the properties of Eigenvalue, we know that the Eigenvalue of an upper triangular matrix is the diagonal element of that matrix.
- This property is also valid for diagonal matrix, scalar matrix, unit matrix, and a lower triangular matrix.
The eigenvalue of Adj A matrix(λ') is given by,
λ' = \(\frac{{\left| A \right|}}{λ }\;\)
The determinant of the upper triangular matrix = λ1 × λ2 × λ3
Augmented matrix is given by, \(\left| {{\bf{A}} - {\bf{\lambda I}}} \right| = 0\), where I is identity matrix.
Calculation:
Given:
\(A = \left( {\begin{array}{*{20}{c}} 2&3&4\\ 0&4&2\\ 0&0&3 \end{array}} \right)\)
Now, we know that
The given matrix is an upper triangular matrix
∴ Eigenvalues of the matrix are
, λ
1 = 2, λ
2 = 4 and λ
3 = 3
And determinant of matrix A =
λ1 × λ2 × λ3 = 2 × 4 × 3 = 24
If λ is an Eigenvalue of a matrix,
then the Eigenvalues of Adj A = \(\frac{{\left| A \right|}}{λ_1 }\;\), \(\frac{{\left| A \right|}}{λ_2}\;\), \(\frac{{\left| A \right|}}{λ_3 }\;\)= \(\frac{24}{2 }, \frac{24}{4 }, \frac{24}{3 } = 12, 6, 8\)
If λ is the Eigenvalue of matrix A, then Augmented matrix is \(\left| {{\bf{A}} - {\bf{\lambda I}}} \right| = 0\) and eigenvalues follows this equation.
i.e. from given equation A 2 - 2A + I ⇒\({λ ^2} - 2λ + 1 ⇒ {\left( {λ - 1} \right)^2}\)
Shortcut:
Put the Eigenvalues 12, 6, and 8 in the above equation, then eigenvalues of A2 - 2A + I are
(12 - 1)2, (6 - 1)2, (8 - 1)2 = 121, 25, 49.