JEE Mathematics 2024 Question with Solution

Given: $B = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$, $AB^{-1} = A^{-1}$, $BCB^{-1} = A$, and $C^4 + \alpha C^2 + \beta I = O$.

Find: $2\beta - \alpha$.

From $AB^{-1} = A^{-1}$, post-multiply by $B$:

Now pre-multiply by $A$:

Using $BCB^{-1} = A$, square both sides:

Hence, multiplying by $B^{-1}$ on the left and by $B$ on the right,

So $C^2$ satisfies the same characteristic equation as $B$. Now,

Replacing $\lambda$ by $C^2$, we get

Comparing with

we obtain

Therefore,

The correct option is D.

Given: $BCB^{-1} = A$ shows that $A$ is similar to $C$.

Find: the value of $2\beta - \alpha$.

From the relation

post-multiplying by $B$ gives

Then pre-multiplying by $A$ gives

Now from similarity,

Squaring both sides,

Since $A^2 = B$,

So,

Thus the polynomial satisfied by $B$ will be satisfied by $C^2$. Compute the characteristic polynomial of $B$:

Hence,

Since $C^2 = B$,

Therefore,

and so

Therefore, the correct option is D.