Question

Complete the following activity to verify A. adj (A) = det (A) I.

Given A = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` then

|A| = 2(____) – 0(____) + ( ) (____)

= 6 – 0 – 5

= ______ ≠ 0

Cofactors of all elements of matrix A are

A₁₁ = `(-1)^2 |("( )", "( )"),("( )", "( )")|` = (______),

A₁₂ = `(-1)^3 |(5, "( )"),("( )", 3)|` = – 15,

A₁₃ = `(-1)^4 |(5, "( )"),("( )", 1)|` = 5,

A₂₁ = _______, A₂₂ = _______, A₂₃ = _______,

A₃₁ = `(-1)^4 |("( )", "( )"),("( )", "( )")|` = (______),

A₃₂ = `(-1)^5 |(2, "( )"),("( )", 0)|` = (  ),

A₃₃ = `(-1)^6 |(2, "( )"),("( )", 1)|` = 2,.

Cofactors of matrix A = `[(3, "____", "____"),("____", "____",-2),(1, "____", "____")]`

adj (A) = `[("____", "____", "____"),("____", "____","____"),("____","____","____")]`

A.adj (A) = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)] [("( )", -1, 1), (-15, "( )", -5),("( )", -2, "( )")] = [(1, 0, "( )"),("( )", "( )", "( )"),(0, "( )", "( )")]` = |A|I

Answer 1

Given A = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` 

|A| = `|(2, 0, -1),(5, 1, 0),(0, 1, 3)|`

= 2(3 – 0) – 0(15 – 0) + (– 1) (5 – 0)

= 6 – 0 – 5

= 1 ≠ 0

Cofactors of all elements of matrix A are

A₁₁ = (–1)² `|(1, 0),(1, 3)|` = 3 – 0 = 3,

A₁₂ = (–1)³ `|(5, 0),(0, 3)|` = – 15,

A₁₃ = (–1)⁴ `|(5, 1),(0, 1)|` = 5,

A₂₁ = (–1)³ `|(0, -1),(1, 3)|`

= (– 1) (0 + 1)

= – 1

A₂₂ = (–1)⁴ `|(2, -1),(0, 3)|`

= 1(6 + 0)

= 6

A₂₃ = (–1)⁵ `|(2, 0),(0, 1)|`

= (–1) (2 – 0)

= – 2

A₃₁ = (–1)⁴ `|(0, -1), (1, 0)|` 

= (1) (0 + 1)

= 1,

A₃₂ = (–1)⁵ `|(2, -1), (5, 0)|`

= (–1) (0 + 5)

= – 5

A₃₃ = (–1)⁶ `|(2, 0), (5, 1)|`

= (1) (2 – 0)

= 2

Cofactors of matrix A = `[(3, -15, 5),(-1, 6, -2),(1, -5, 2)]`,

adj (A) = `[(3, -15, 5),(-1, 6, -2),(1, -5, 2)]^"T"`

= `[(3, -1, 1),(-15, 6, -5),(5, -2, 2)]`

A.adj (A) = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)] [(3, -1, 1), (-15, 6, -5),(5, -2, 2)]`

= `[(6 - 0 - 5, -2 + 0 + 2, 2 - 0 - 2),(15 - 15 + 0, -5 + 6 - 0, 5 - 5+ 0),(0 - 15 + 15, 0 + 6 - 6, 0 - 5 + 6)]`

= `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

= |A|I