JEE Main 2024 — 29 January, Shift 2 — Question 1

To find the sum of the prime factors of $|P^{-1}AP - 2I|$, we first utilize a property of determinants related to matrix similarity.

We are given the matrices: $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ $P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$

The expression we need to evaluate is $|P^{-1}AP - 2I|$. We can manipulate the expression inside the determinant: $P^{-1}AP - 2I = P^{-1}AP - P^{-1}(2I)P$ (since $P^{-1}(2I)P = 2P^{-1}IP = 2P^{-1}P = 2I$) Factor out $P^{-1}$ from the left and $P$ from the right: $P^{-1}AP - 2I = P^{-1}(A - 2I)P$

Now, we can take the determinant of this expression: $|P^{-1}(A - 2I)P| = |P^{-1}| |A - 2I| |P|$ Since the determinant of an inverse matrix is the reciprocal of the determinant of the original matrix, i.e., $|P^{-1}| = \frac{1}{|P|}$, we have $|P^{-1}| |P| = 1$. Therefore, the expression simplifies to: $|P^{-1}(A - 2I)P| = |A - 2I|$

So, the problem reduces to calculating the determinant of the matrix $(A - 2I)$. First, form the matrix $A - 2I$: $A - 2I = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix} - 2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ $A - 2I = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix} - \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$ $A - 2I = \begin{bmatrix} 2-2 & 1-0 & 2-0 \\ 6-0 & 2-2 & 11-0 \\ 3-0 & 3-0 & 2-2 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 2 \\ 6 & 0 & 11 \\ 3 & 3 & 0 \end{bmatrix}$

Now, calculate the determinant of this resulting matrix: $|A - 2I| = 0 \cdot \begin{vmatrix} 0 & 11 \\ 3 & 0 \end{vmatrix} - 1 \cdot \begin{vmatrix} 6 & 11 \\ 3 & 0 \end{vmatrix} + 2 \cdot \begin{vmatrix} 6 & 0 \\ 3 & 3 \end{vmatrix}$ $= 0 - 1((6)(0) - (11)(3)) + 2((6)(3) - (0)(3))$ $= -1(0 - 33) + 2(18 - 0)$ $= -1(-33) + 2(18)$ $= 33 + 36$ $= 69$

The value of $|P^{-1}AP - 2I|$ is 69. Finally, we need to find the sum of the prime factors of 69. The prime factorization of 69 is $3 \times 23$. The prime factors are 3 and 23. The sum of the prime factors is $3 + 23 = 26$.

The final answer is $\boxed{26}$.