QJEEM21Ma3a73: JEE Main 2021 Mathematics - Class 12 JEEM21Ma3a73 Solution: Let A and B be 3 × 3 real matrices such that A is symmetric…

Question

Verified Solution • 2025–26 Session moderate

Let A and B be 3

\times

3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A²B²

-

B²A²) X = O, where X is a 3

\times

1 column matrix of unknown variables and O is a 3

\times

1 null matrix, has :

Accepted Answer

A^T = A, B^T =

-

B

Let A²B²

-

B²A² = P

P^T = (A²B²

-

B²A²)^T = (A²B²)^T

-

(B²A²)^T

= (B²)^T (A²)^T

-

(A²)^T (B²)^T

= B²A²

-

A²B²

\Rightarrow

P is skew-symmetric matrix

[\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]

∴

ay + bz = 0 ..... (1)

-

ax + cz = 0 .... (2)

-

bx

-

cy = 0 ..... (3)

From equation 1, 2, 3

Δ

= 0 &

Δ

₁ =

Δ

₂ =

Δ

₃ = 0

∴

equation have infinite number of solution

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