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

Concept:

Characteristics polynomial of a matrix:

Let A be a square matrix of order n and λ be any scalar quantity. Then the polynomial in λ of degree n formed by solving the characteristic equation |A - λI| = 0 is called the characteristics polynomial of the matrix.

Eigenvalues:

The roots of the characteristic equation are called the eigenvalues or characteristic roots of latent roots of matrix A.

Eigenvectors:

If λ is the eigenvalue of matrix A then a non-zero vector X that satisfies AX = λX is called the eigenvector of the matrix corresponding to the eigenvalue λ.

Calculation:

\(\left[ {\begin{array}{*{20}{c}} {\left( {1 - \lambda } \right)}&2\\ 0&{\left( {2 - \lambda } \right)} \end{array}} \right] =0\)

\(\Rightarrow \left( {1 - \lambda } \right)\left( {2 - \lambda } \right) = 0\)

∴ λ = 1, 2

Putting the value of λ = 1

⇒ \(\left[ {\begin{array}{*{20}{c}} 0&2\\ 0&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ a \end{array}} \right] = 0\)

⇒ a = 0

Putting λ = 2,

\(\left[ {\begin{array}{*{20}{c}} { - 1}&2\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1\\ b \end{array}} \right] = 0\)

⇒ \({\rm{b\;}} = \frac{1}{2}{\rm{\;}}\)

∴ \({\rm{a\;}} + {\rm{\;b\;}} = \frac{1}{2}{\rm{\;}}\)