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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
Solution
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
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