Class 12 Determinants · Long Answer3 Marksorthogonal_matrices

Class 12 Maths · Determinants · Long Answer #55

Find all real x,y,z such that A=\(\begin{bmatrix}0 & 2y & z\\x & y & −z\\x & −y & z\end{bmatrix}\) satisfies Aᵀ=A⁻¹.

✅ Step-by-Step Solution

Final Answer: x=±1/√2,y=±1/√6,z=±1/√3, with independent signs.

The condition is AAᵀ=I. Its off-diagonal entries include 2y²−z² and x²−y²−z²; its first diagonal entry is 4y²+z².
Equating to I gives 2y²=z² and 4y²+z²=1. Hence 6y²=1, so y²=1/6 and z²=1/3.
The second diagonal equation x²+y²+z²=1 gives x²=1/2. These squared values make all diagonal entries 1 and all off-diagonal entries 0, independently of the signs.
Thus x=±1/√2,y=±1/√6,z=±1/√3, with all eight independent sign choices valid.

[0.5 marks] The condition is AAᵀ=I. Its off-diagonal entries include 2y²−z² and x²−y²−z²; its first diagonal entry is 4y²+z².
[1 mark] Equating to I gives 2y²=z² and 4y²+z²=1. Hence 6y²=1, so y²=1/6 and z²=1/3.
[1 mark] The second diagonal equation x²+y²+z²=1 gives x²=1/2. These squared values make all diagonal entries 1 and all off-diagonal entries 0, independently of the signs.
[0.5 marks] Thus x=±1/√2,y=±1/√6,z=±1/√3, with all eight independent sign choices valid.

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