The characteristic equation for the matrix $A = \left( \begin{array} { c c c }
3 & 0 & 0 \
0 & 1 & - 1 \
0 & 1 & 3
\end{array} \right)$ is
A) $- \lambda ^ { 3 } - 7 \lambda ^ { 2 } - 16 \lambda + 12 = 0$
B) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } + 14 \lambda + 6 = 0$
C) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 14 \lambda + 6 = 0$
D) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } + 16 \lambda + 12 = 0$
E) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 16 \lambda + 12 = 0$

Question

The characteristic equation for the matrix $A = \left( \begin{array} { c c c } 
3 & 0 & 0 \
0 & 1 & - 1 \
0 & 1 & 3
\end{array} \right)$ is
A) $- \lambda ^ { 3 } - 7 \lambda ^ { 2 } - 16 \lambda + 12 = 0$
B) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } + 14 \lambda + 6 = 0$
C) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 14 \lambda + 6 = 0$
D) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } + 16 \lambda + 12 = 0$
E) $- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 16 \lambda + 12 = 0$

Quizplus · Accepted Answer

The characteristic equation of a matrix $A$ is found by calculating the determinant of $A - \lambda I$, where $I$ is the identity matrix and $\lambda$ is a scalar. For the given matrix $A$, the characteristic equation is obtained by calculating the determinant of $A - \lambda I = \left( \begin{array}{ccc} 3-\lambda & 0 & 0 \ 0 & 1-\lambda & -1 \ 0 & 1 & 3-\lambda \end{array} \right)$. This results in a polynomial $-\lambda^3 + 7\lambda^2 - 16\lambda + 12 = 0$, matching option E.

The Characteristic Equation for the Matrix A=(30001−1013)A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 1 & - 1 \\0 & 1 & 3\end{array} \right)A=​300​011​0−13​​

The Characteristic Equation for the Matrix $A = \left( \begin{array} { c c c } 3 & 0 & 0 \\0 & 1 & - 1 \\0 & 1 & 3\end{array} \right)$