Maths Q4 – JEE-MAIN 2026

Let

A = [\begin{matrix} 1 & 2 1 & α \end{matrix}]

and

B = [\begin{matrix} 3 & 3 β & 2 \end{matrix}]

. If

A^{2} - 4 A + I = O

and

B^{2} - 5 B - 6 I = O

, then among the two statements : (S1):

[(B - A) (B + A)]^{T} = [\begin{matrix} 13 & 15 7 & 10 \end{matrix}]

and (S2):

det (adj (A + B)) = - 5

The given matrix equations for $A$ and $B$ are their characteristic equations.

● Video Walkthrough

Step 1: Determine

α

and

β

✦ Active

The characteristic equation for a $2 \times 2$ matrix $M$ is $M^{2} - Tr (M) M + det (M) I = O$ .

For

A^{2} - 4 A + I = O

Tr (A) = 1 + α = 4 ⟹ α = 3

det (A) = 1 \cdot α - 2 \cdot 1 = α - 2 = 1 ⟹ α = 3

. So,

A = [\begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix}]

. For

B^{2} - 5 B - 6 I = O

Tr (B) = 3 + 2 = 5

det (B) = 3 \cdot 2 - 3 \cdot β = 6 - 3 β = - 6 ⟹ 3 β = 12 ⟹ β = 4

. So,

B = [\begin{matrix} 3 & 3 \\ 4 & 2 \end{matrix}]

Step 2: Evaluate Statement (S1)○ Expand

First, calculate $B - A$ and $B + A$ :

B - A = [\begin{matrix} 3 & 3 \\ 4 & 2 \end{matrix}] - [\begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix}] = [\begin{matrix} 2 & 1 \\ 3 & - 1 \end{matrix}]

B + A = [\begin{matrix} 3 & 3 \\ 4 & 2 \end{matrix}] + [\begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix}] = [\begin{matrix} 4 & 5 \\ 5 & 5 \end{matrix}]

. Now, calculate

(B - A) (B + A)

(B - A) (B + A) = [\begin{matrix} 2 & 1 \\ 3 & - 1 \end{matrix}] [\begin{matrix} 4 & 5 \\ 5 & 5 \end{matrix}] = [\begin{matrix} (2) (4) + (1) (5) & (2) (5) + (1) (5) \\ (3) (4) + (- 1) (5) & (3) (5) + (- 1) (5) \end{matrix}] = [\begin{matrix} 13 & 15 \\ 7 & 10 \end{matrix}]

. Finally, calculate

[(B - A) (B + A)]^{T}

[(B - A) (B + A)]^{T} = [\begin{matrix} 13 & 7 \\ 15 & 10 \end{matrix}]

. Since

[\begin{matrix} 13 & 7 \\ 15 & 10 \end{matrix}] \neq [\begin{matrix} 13 & 15 \\ 7 & 10 \end{matrix}]

, statement (S1) is incorrect.

Step 3: Evaluate Statement (S2)○ Expand

First, calculate $A + B$ :

A + B = [\begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix}] + [\begin{matrix} 3 & 3 \\ 4 & 2 \end{matrix}] = [\begin{matrix} 4 & 5 \\ 5 & 5 \end{matrix}]

. Next, calculate

det (A + B)

det (A + B) = (4) (5) - (5) (5) = 20 - 25 = - 5

. For a

2 \times 2

matrix

M

, the determinant of its adjoint is given by

det (adj (M)) = (det (M))^{2 - 1} = det (M)

. Therefore,

det (adj (A + B)) = det (A + B) = - 5

. Since the calculated value is

- 5

, statement (S2) is correct. Based on the evaluation, (S1) is incorrect and (S2) is correct. Thus, only (S2) is correct.

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Maths Question 4 – JEE-MAIN 2026