Let $I = I _ { 2 }$ be the identity matrix of order 2, and let $f ( x ) = | A - x I |$ . Find the polynomial $f ( x )$ for the given matrix A in order to find the zeros of $f ( x )$ .
(In the study of matrices, $f ( x )$ is the characteristic polynomial of A, and the zeros of $f ( x )$ are the characteristic values (eigenvalues) of A.) $$A = \left[ \begin{array} { r r }
37 & - 90 \
12 & - 29
\end{array} \right]$$
A) $x = 7 , x = 1$
B) $x = - 7 , x = 1$
C) $x = 4 , x = 5$
D) $x = - 10 , x = 6$
E) $x = 7 , x = 4$

Question

Let $I = I _ { 2 }$ be the identity matrix of order 2, and let $f ( x ) = | A - x I |$ . Find the polynomial $f ( x )$ for the given matrix A in order to find the zeros of $f ( x )$ .
(In the study of matrices, $f ( x )$ is the characteristic polynomial of A, and the zeros of $f ( x )$ are the characteristic values (eigenvalues) of A.) $$A = \left[ \begin{array} { r r } 
37 & - 90 \
12 & - 29
\end{array} \right]$$
A) $x = 7 , x = 1$
B) $x = - 7 , x = 1$
C) $x = 4 , x = 5$
D) $x = - 10 , x = 6$
E) $x = 7 , x = 4$

Quizplus · Accepted Answer

The characteristic polynomial is $$f(x) = |A - xI| = \left| \begin{array}{cc} 37-x & -90 \\ 12 & -29-x \end{array} \right| = (37-x)(-29-x) - (-90)(12) = x^2 - 6x + 1073.$$The zeros of $f(x)$ are the eigenvalues of $A$, which are $x = 7$ and $x = 1$.

Let I=I2I = I _ { 2 }I=I2​ Be the Identity Matrix of Order 2, and Let

Let $I = I _ { 2 }$ Be the Identity Matrix of Order 2, and Let