If A is a 2 × 2 matrix and I is an identity matrix of order 2 & A − λI = 0 gives value of λ as –1 & 3 then trace of A2 is equal to ______

The correct answer is 10.

Step-by-Step Explanation:

1. Understanding Eigenvalues:
The given equation $|A-\lambda I|=0$ is the characteristic equation of the matrix $A$. The roots of this equation, represented by $\lambda$, are the eigenvalues of matrix $A$.
Here, the eigenvalues of the 2 × 2 matrix $A$ are given as ${\lambda }_1=-1$ and ${\lambda }_2=3$.

2. Eigenvalues of $A^2$:
According to the properties of matrix eigenvalues, if $\lambda$ is an eigenvalue of a matrix $A$, then ${\lambda }^n$ is an eigenvalue of the matrix $A^n$.
Therefore, the eigenvalues of the matrix $A^2$ are:
${\lambda }_1^{\prime }={(-1)}^2=1$
${\lambda }_2^{\prime }={\left(3\right)}^2=9$

3. Calculating the Trace of $A^2$:
The trace of a matrix is defined as the sum of its diagonal elements, which is also mathematically equal to the sum of all its eigenvalues.
Thus, the trace of matrix $A^2$ is the sum of its eigenvalues:
$\text{Trace}\left(A^2\right)={\lambda }_1^{\prime }+{\lambda }_2^{\prime }$
$\text{Trace}\left(A^2\right)=1+9=10$