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NYC.L11

# NYC.L11 - MAT-NYC 1 Let A = LAB#11 Determinants 1 2 −2 3...

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Unformatted text preview: MAT-NYC 1. Let A = LAB #11 Determinants 1 2 −2 3 3 −1 50 4 0 21 1 7 21 . Find the cofactors c31 , c23 , c42 . 2. Compute the determinant 0 −7 3 −5 0 2 0 0 3 −6 4 −8 0 5 2 −3 0 9 −1 2 4 0 7 5 0 . 3. Let A be a 4 × 4 matrix with det A = a. Find an expression for det B if B is the matrix obtained from A by the following sequence of operations: R3 ← (1/2)R3 , R2 ← R2 −R1 , R1 ↔ R2 and R3 ← 3R3 − 5R2 .(Note that the last operation is not elementary.) abc −x −y −z −2a −2b −2c 4. If p q r = −1, compute: a) 3p + a 3q + b 3r + c ; b) 2p + x 2q + y 2r + z xyz 2p 2q 2r 3x 3y 3z t−2 4 3 t + 1 −2 is singular. For each of 5. Find for what values of t the matrix 1 0 0 t−4 these values, ﬁnd the rank of the matrix. 6. a) Evaluate by inspection b) Show that 1 a a2 1 b b2 1 c c2 a b c a + b 2b c + b . 2 2 2 = (b − a)(c − a)(c − b) . (This is a Vandermonde determi- nant.) 7. Evaluate the following determinants by combining row or column operations with cofactor expansions. Be eﬃcient. a) 1 2 2 3 1 0 −2 0 3 −1 1 −2 4 −3 0 2 b) 3 1 3 2 3 0 1 −2 1 −1 4 3 2 2 −1 1 c) 3 0 1 2 3 5 −2 −2 6 0 −4 00 2 4 −1 1 4 1 15 7 0 53 / ... MAT-NYC LAB #11, Page 2 8. Determine which of the following statements are true or false. a) The determinant det A is deﬁned for any matrix A. b) The determinant det A is deﬁned for each square matrix A. c) If two rows and also two columns of a square matrix A are interchanged, the determinant changes sign. d) The determinant of an elementary matrix is nonzero. e) If A is a 2 × 2 matrix with zero determinant, then one column of A must be a multiple of the other. f) If A is a 3 × 3 matrix with zero determinant, then two of the rows of A must be proportional. Answers: 1. c31 = −103, c23 = −71, c42 = −31. 2. 6. 3. −3a/2 4. a) 2; b) 12 5. t = −2, 3, 4. For each of these values, the rank of the matrix is 2. 6. a) 0 7. a) −131; b) −34; c) −72. 8. a) F; b) T; c) F; d) T; e) T; f) F. ...
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NYC.L11 - MAT-NYC 1 Let A = LAB#11 Determinants 1 2 −2 3...

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