Let A and B be square matrices of order 4 such that AAT = ATA and B = (A3 – (AT)3 – 3A2AT + 3A(AT)2)11. If B = adj. (B + 3BT) and det (B) > 0 then[Note: adj(P) denotes adjoint matrix of matrix P.]

Engineering

Mathematics

Transpose and Adjoint of a Matrix

Question

Let A and B be square matrices of order 4 such that AA^T = A^TA and B = (A³ – (A^T)³ – 3A²A^T + 3A(A^T)²)¹¹. If B = adj. (B + 3B^T) and det (B) > 0 then

[Note: adj(P) denotes adjoint matrix of matrix P.]

B^–1 = – 8B

$det (B) = \frac{1}{16}$

$det (B) = \frac{1}{64}$

B^–1 = – 4B

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B = (A – A^T)³³

Þ B is skew symmetric matrix of order 4.

B = adj(B + 3B^T) = adj(B – 3B) = adj (– 2B)

⇒ | B | = | – 2B |³

⇒ B | = (– 2)¹² | B |³

$\Rightarrow {| B |}^{2} = \frac{1}{2^{12}} \Rightarrow | B | = \frac{1}{64}$

Again B = adj(– 2B)

⇒ B = (– 2)³adj. B

⇒ B = (– 8) | B | B^–1

⇒ B^–1 = – 8B.