Find x, y, z if A = [(0,2y,z),(x,y,-z),(x,-y,z)] satisfies A’ = A^–1

Question

Find x, y, z if A = \(\begin{bmatrix} 0 & 2y & z \\[0.3em] x & y & -z \\[0.3em] x & -y & z\end{bmatrix}\) satisfies A’ = A^–1.

Answer 1

Given,

We need to find x, y and z such that A’ = A^-1

If A’ = A^-1

Pre-multiplying A on both sides:

AA’ = AA^-1

⇒ AA’ = I where I is the identity matrix.

On equating the corresponding elements of matrix.

We need basically 3 equations as we have 3 variables to solve for. You can pick any three elements and equate them.

We have:

4y² + z² = 1 …(1)

x² + y² + z² = 1 …(2)

2y² – z² = 0 …(3)

Adding equation 2 and 3:

6y² = 1

⇒ y² = 1/6

From equation 3:

Z² = 2y²

⇒ z² = 2(1/6)

∴ z² = 1/3

∴ 

From equation 2:

x² = 1 – y² – z²

⇒ x² = 1 – (1/6) – (1/3)

⇒ x² = 1 – 1/2 = 1/2