Question

Let A = `[(i, -i),(-i, i)], i = sqrt(-1)`. Then, the system of linear equations `A^8[(x),(y)] = [(8),(64)]` has ______.

पर्याय

• No solution

• Exactly two solutions

• A unique solution

• Infinitely many solutions

Answer 1

Let A = `[(i, -i),(-i, i)], i = sqrt(-1)`. Then, the system of linear equations `A^8[(x),(y)] = [(8),(64)]` has no solution.

Explanation:

Given A = `[(i, -i),(-i, i)], i = sqrt(-1)`  ...(i)

We have to find the solution of the system of linear equations

`A^8[(x),(y)] = [(8),(64)]`  ...(ii)

Now using equation (i)

A² = `[(i, -i),(-i, i)][(i, -i),(-i, i)]`

⇒ A² = `[(i^2 + i^2, -i^2 - i^2),(-i^2 - i^2, i^2 + i^2)]`  ...(∵ i² = –1)

⇒ A² = `[(-2, 2),(2, -2)]`

⇒ A² = `2[(-1, 1),(1, -1)]`

⇒ A⁴ = `4[(-1, 1),(1, -1)][(-1, 1),(1, -1)]`

⇒ A⁴ = `4[(2, -2),(-2, 2)]`

⇒ A⁴ = `8[(1, -1),(-1, 1)]`

⇒ A⁸ = `64[(1, -1),(-1, 1)][(1, -1),(-1, 1)]`

⇒ A⁸ = `64[(2, -2),(-2, 2)]`

⇒ A⁸ = `128[(1, -1),(-1, 1)]`  ...(iii)

Now using the equation (ii) and (iii)

`128[(1, -1),(-1, 1)][(x),(y)] = [(8),(64)]`

⇒ `128[(x - y),(-x + y)] = [(8),(64)]`

⇒ `[(x - y),(-x + y)] = 1/128[(8),(64)]`

x – y = `1/16`  ...(iv)

–x + y = `1/2`  ...(v)

From equation (iv) and (v)

System of equation has no solution.