JEE Mathematics 2024 Question with Solution

Given: $\Lambda$ is a $2 \times 2$ real matrix and the roots of $|\Lambda - xI| = 0$ are $-1$ and $3$.

Find: The sum of the diagonal elements of $\Lambda^2$.

The roots of $|\Lambda - xI| = 0$ are the eigenvalues of $\Lambda$. Hence the eigenvalues are $-1$ and $3$.

For $\Lambda^2$, the sum of the diagonal elements equals $\text{tr}(\Lambda^2)$, and this is the sum of the squares of the eigenvalues of $\Lambda$.

Therefore, the sum of the diagonal elements of $\Lambda^2$ is $10$. Hence, the correct option is A.

Given: Let

with eigenvalues $-1$ and $3$.

Find: $\text{tr}(\Lambda^2)$.

From the eigenvalues,

and

Now,

So the sum of the diagonal elements is

Using

we get

Since

substitute into the previous expression:

Therefore,

Thus, the sum of the diagonal elements of $\Lambda^2$ is $10$.