Question

If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.

पर्याय

• `[(x^(-1),0,0),(0, y^(-1),0),(0,0,z^(-1))]`

• `xyz[(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1))]`

• `1/(xyz)[(x,0,0),(0,y,0),(0,0,z)]`

• `1/(xyz)[(1,0,0),(0,1,0),(0,0,1)]`

Answer 1

If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is `underlinebb([(x^(-1),0,0),(0, y^(-1),0),(0,0,z^(-1))])`.

Explanation:

Let, A = `[(x,0,0),(0,y,0),(0,0,z)]`

∴ |A| = x[yz − 0] = xyz

∴ A₁₁ = `(-1)^(1 + 1)[(y,0),(0,z)]`

= (−1)² [yz − 0]

= 1 × yz

= yz

A₁₂ = `(-1)^(1 + 2)[(0,0),(0,z)]`

= (−1)³ [0 − 0]

= 0

A₁₃ = `(-1)^(1 + 3)[(0,y),(0,0)]`

= (−1)⁴ [0 − 0]

= 0

A₂₁ = `(-1)^(2 + 1)[(0,0),(0,z)]`

= (−1)³ [0 − 0]

= 0

A₂₂ = `(-1)^(2 + 2)[(x,0),(0,z)]`

= (−1)⁴ [xz − 0]

= 1 × zx

= zx

A₂₃ = `(-1)^(2 + 3)[(x,0),(0,0)]`

= (−1)⁵ [0 − 0]

= 0

A₃₁ = `(-1)^(3 + 1)[(0,0),(0,z)]`

= (−1)⁴ [0 − 0]

= 0

A₃₂ = `(-1)^(3 + 2)[(x,0),(0,0)]`

= (−1)⁵ [0 − 0]

= 0

A₃₃ = `(-1)^(3 + 3)[(x,0),(0,y)]`

= (−1)⁶ [xy − 0]

= xy

∴ adj A = `[(yz,0,0),(0,zx,0),(0,0,xy)] = [(yz,0,0),(0,zx,0),(0,0,xy)]`

A⁻¹ = `1/|A|` (adj A)

= `1/(xyz)[(yz,0,0),(0,zx,0),(0,0,xy)]`

= `[(1/x,0,0),(0,1/y,0),(0,0,1/z)]`

= `[(x^-1,0,0),(0,y^-1,0),(0,0,z^-1)]`