Lecture 23 - ENGR-1100 Introduction to Engineering Analysis...

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Lecture 23 Notes courtesy of: Prof. Yoav Peles ENGR-1100 Introduction to Engineering Analysis
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Homework Assignment (due next class) L.A.S. Set 2.3-3 (page 89) L.A.S. Set 2.3-7 (page 89) L.A.S. Set 2.3-10 (page 89)
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Lecture outline Adjoint formula for A -1 Cramer’s rule.
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Definition If A is any nxn matrix and C ij is the cofactor of a ij , then the matrix C 11 C 12….… C 1n C 21 C 22….… C 2n : : : : : : C n1 C n2….… C nn is called the matrix of cofactors from A . The transpose of this matrix is called the adjoint of A and is denoted by adj (A) .
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Example 1 1 0 1 -1 3 0 1 0 2 A= The cofactors of A are C 11 =6; C 12 =2; C 13 =-3 C 21 =0 ; C 22 =1; C 23 =0 31 32 33 It follows that the matrix of cofactors of A is 6 2 -3 0 1 0 -3 -1 3
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And the adjoint of A is: matrix of cofactors of A Adj(A)= 6 0 -3 2 1 -1 -3 0 3 6 2 -3 0 1 0 -3 -1 3
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Theorem 1 If A is an invertible matrix, then A -1 = adj(A) det(A) 1
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Example 2 Use theorem 1 to find the inverse of the matrix A in the previous example 1 0 1 -1 3 0 1 0 2 A= Adj(A)= 6 0 -3 2 1 -1 -3 0 3 det(A)= =1*3*2-1*3*1=3
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A -1 = adj(A) det(A) 1 And since Adj(A)= 6 0 -3 2 1 -1 -3 0 3 det(A)=3 A -1 = 2 0 -1 2/3 1/3 -1/3 -1 0 1 It follows:
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Lecture 23 - ENGR-1100 Introduction to Engineering Analysis...

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