Let \(\rm \begin{pmatrix}2&a\\\ b&c\end{pmatrix}\) be a 2 x 2 real matrix for which 6 is an eigenvalue. Which of the following statements is necessarily true?

Concept:

Characteristic Equation:

The characteristic equation of a \(2 \times 2\) matrix \(A\) is obtained by calculating the determinant of \((A - \lambda I)\).

For the matrix \(A = \begin{pmatrix} 2 & a \\ b & c \end{pmatrix}\), the characteristic equation is
\(\det(A - \lambda I) = \det \begin{pmatrix} 2 - \lambda & a \\ b & c - \lambda \end{pmatrix}\)

Explanation:

\(\begin{pmatrix} 2 & a \\ b & c \end{pmatrix}\) It is stated that 6 is an eigenvalue of this matrix. To determine which of the statements is necessarily true,

we will compute the characteristic equation of the matrix and relate it to the given eigenvalue.

The characteristic polynomial of a 2x2 matrix is given by:

Det (\(\begin{pmatrix} 2 & a \\ b & c \end{pmatrix}\)- \(\lambda\)\(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)) 

The determinant is

\((2 - \lambda)(c - \lambda) - ab \)

This simplifies to

\((2 - \lambda)(c - \lambda) - ab = \lambda^2 - (2 + c)\lambda + (2c - ab)\)

Since 6 is an eigenvalue, the characteristic polynomial must have \( \lambda = 6 \) as a root.

Substituting \( \lambda = 6 \) into the characteristic equation:

\(6^2 - (2 + c)6 + (2c - ab) = 0 \)

⇒\(36 - 6(2 + c) + (2c - ab) = 0\)

⇒ \(36 - 12 - 6c + 2c - ab = 0\)

⇒ \(24 - 4c - ab = 0\)

Thus, the equation becomes

\(24 - ab = 4c\)

This matches the first statement provided \(24 - ab = 4c\)

Hence, option 1) is correct.