Lect9 - Chapter 2. Determinants Math1111 Adjoint of a...

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Unformatted text preview: Chapter 2. Determinants Math1111 Adjoint of a Matrix Definition Definition Let A be a square matrix of order n . The adjoint of A , denoted by adj A , is a matrix defined as adj A = A 11 A 21 ··· A n 1 A 12 A 22 ··· A n 2 . . . . . . . . . A 1 n A 2 n ··· A nn where A ij is the cofactor of a ij . Chapter 2. Determinants Math1111 Adjoint of a Matrix Theorem Theorem A ( adj A ) = ( det A ) I . Chapter 2. Determinants Math1111 Adjoint of a Matrix Theorem Theorem A ( adj A ) = ( det A ) I . Proof. The ( i , j ) th entry of A ( adj A ) is a i 1 A j 1 + a i 2 A j 2 + ··· + a in A jn = det A if i = j , if i 6 = j . Chapter 2. Determinants Math1111 Adjoint of a Matrix Theorem Theorem A ( adj A ) = ( det A ) I . Proof. The ( i , j ) th entry of A ( adj A ) is a i 1 A j 1 + a i 2 A j 2 + ··· + a in A jn = det A if i = j , if i 6 = j . Corollary If det A 6 = , then A- 1 = 1 det A adj A . Chapter 2. Determinants Math1111 Determinants Homework 4 Reading Leon (7th edition): p.106 Leon (8th edition): p.98 Homework 4 Leon (7th edition): Chapter 2 - Section 3 Qn. 1, 5, 6, 8-13 . Leon (8th edition): Section 2.3 Qn. 1, 5, 6, 8-13 . Further Reading Leon (7th edition): p.107 - p.109. Leon (8th edition): p.100 - p.101, p.102-103 (Cross product). Chapter 3. Vector Spaces Math1111 Vector Spaces Motivation Recall that we call a column matrix u = x 1 x 2 . . . x n a vector. Chapter 3. Vector Spaces Math1111 Vector Spaces Motivation Recall that we call a column matrix u = x 1 x 2 . . . x n a vector. Why did we call u a vector? When n = 2 , we associate u to an “arrow” (the vector we learnt in physics). Chapter 3. Vector Spaces Math1111 Vector Spaces Motivation Recall that we call a column matrix u = x 1 x 2 . . . x n a vector. Why did we call u a vector? When n = 2 , we associate u to an “arrow” (the vector we learnt in physics). For these arrows, there are two operations defined on them. Chapter 3. Vector Spaces Math1111 Vector Spaces Motivation Recall that we call a column matrix u = x 1 x 2 . . . x n a vector. Why did we call u a vector? When n = 2 , we associate u to an “arrow” (the vector we learnt in physics)....
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Lect9 - Chapter 2. Determinants Math1111 Adjoint of a...

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