Let the Eigenvector of the matrix \(\left[ {\begin{array}{*{20}{c}} 1&2\\ 0&2 \end{array}} \right]\) be 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 the value of (a + b)?

Concept:

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 Eigenvalues or latent roots or characteristic roots of matrix A.

Eigenvector (X) that corresponding to Eigenvalue (λ) satisfies the equation AX = λX.

Properties of Eigenvalues:

The sum of Eigenvalues of a matrix A is equal to the trace of that matrix A

The product of Eigenvalues of a matrix A is equal to the determinant of that matrix A

Calculation:

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

|A – λI| = 0

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

⇒ (1 – λ) (2 – λ) = 0

⇒ λ = 1, 2

Given Eigenvectors are: \(\left[ {\begin{array}{*{20}{c}} 1\\ a \end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}} 1\\ b \end{array}} \right]\)

For λ = 1, let the Eigenvector is \(\left[ {\begin{array}{*{20}{c}} 1\\ a \end{array}} \right]\)

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

\( \Rightarrow \left[ {\begin{array}{*{20}{c}} {1 + 2a}\\ {2a} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1\\ a \end{array}} \right]\)

⇒ a = 0

For λ = 2, let the Eigenvector is \(\left[ {\begin{array}{*{20}{c}} 1\\ b \end{array}} \right]\)

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

\( \Rightarrow \left[ {\begin{array}{*{20}{c}} {1 + 2b}\\ {2b} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2\\ {2b} \end{array}} \right]\)

⇒ b = 1/2

Now, a + b = 1/2