Question

Characteristic equation of the matrix $A = \begin{bmatrix} 1 & 1 & 3 \ 1 & 3 & - 3 \

• 2 & - 4 & - 4 \end{bmatrix}$

• A. $A^{3} - 20A + 8I$
• B. $A^{3} + 20A + 8I$
• C. $A^{3} - 80A + 20I$
• D. None of these

Answer 1

Answer: (A)

$A^{3} - 20A + 8I$

Answer 2

The characteristic equation is $|A - \lambda I| = 0$.

So, $\left| \begin{matrix} 1 - \lambda & 1 & 3 \ 1 & 3 - \lambda & - 3 \

• 2 & - 4 & - 4 - \lambda \end{matrix} \right| = 0$i.e.$\lambda^{3} - 20\lambda + 8 = 0$

By cayley-Hamilton theorem , $A^{3} - 20A + 8I = 0$