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(a) (b) (c) Let A be an nxn matrix and cij is the corresponding cofactor of Aij. Starting from the fact that if two rows of A are identical, the determinant [det(4) ifi = j 0 if i = j Following (a), if we define a new matrix called the adjoint of A by of A is 0. Prove 4₁€ ₁₁ + A₁₂ € ₁2 + 0. A = A1 A2 A21 A22 : : n1 ... + Ain C jn A A2n : nn Please show that A(adj A) = det(A)I,. = adj A = Following (b), please prove that A¹ = C11 €21 C12 €22 : Cin 1 det (4) C2n C₂1 Cn2 : Cnn (adj A) whenever det(4) #

(a) (b) (c) Let A be an nxn matrix and cij is the corresponding cofactor of Aij. Starting from the fact that if two rows of A are identical, the determinant [det(4) ifi = j 0 if i = j Following (a), if we define a new matrix called the adjoint of A by of A is 0. Prove 4₁€ ₁₁ + A₁₂ € ₁2 + 0. A = A1 A2 A21 A22 : : n1 ... + Ain C jn A A2n : nn Please show that A(adj A) = det(A)I,. = adj A = Following (b), please prove that A¹ = C11 €21 C12 €22 : Cin 1 det (4) C2n C₂1 Cn2 : Cnn (adj A) whenever det(4) #

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. of A is 0. Prove A₁₁₁+A₁₂₁2 + + AinCjn = { j2 0 (b) (c) Let A be an nxn matrix and cj is the corresponding cofactor of Aij. Starting from the fact that if two rows of A are identical, the determinant det(A) ifi = j if i #j Following (a), if we define a new matrix called the adjoint of A by (d) 12 €22 4-659 adj A = Cin Can 0. A = A₁ A12 A21 A22 Please show that A(adj A) = det(A)I„ . Following please prove that A¹ = C11 21 1 det(4) Cn2 (adj A) whenever det(4) # 2 Using (c), please calculate the inverse matrix of A = 3 1 2 2 2 1 2 3
Transcribed Image Text:1. of A is 0. Prove A₁₁₁+A₁₂₁2 + + AinCjn = { j2 0 (b) (c) Let A be an nxn matrix and cj is the corresponding cofactor of Aij. Starting from the fact that if two rows of A are identical, the determinant det(A) ifi = j if i #j Following (a), if we define a new matrix called the adjoint of A by (d) 12 €22 4-659 adj A = Cin Can 0. A = A₁ A12 A21 A22 Please show that A(adj A) = det(A)I„ . Following please prove that A¹ = C11 21 1 det(4) Cn2 (adj A) whenever det(4) # 2 Using (c), please calculate the inverse matrix of A = 3 1 2 2 2 1 2 3
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