1. Let A and B be n xn matrices such that det A ±0 and det B 70. Prove that ((AB)")-1 = (A-1)"(B-1)". Hint: Use the properties of the inverse and transpose. 2. Derive an analytical expression for the line which passes through a point with coordinates (a1, a2, a3) and which is perpendicular to vectors b = (b1, b2, b3)" and c = (c1, c2, C3)". %3D 1.2.4 Prove that vectors (1, 1, 1)", (2, 1,0)", (3, 1, 4)", and (1,2, –2)" are linearly dependent. 1.2.5 If a, b, and c are linearly independent vectors in R", prove that a + b, b+c, and a+c are also linearly independent. Is the same true of a – b, b+ c, and a + c? 1.3.3 Give an example where rank(AB) + rank(BA). (Hint: Try some 2 x 2 matrices.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Can you help with question 2?

1. Let A and B be n xn matrices such that det A ±0 and det B 70. Prove that ((AB)")-1 =
(A-1)"(B-1)". Hint: Use the properties of the inverse and transpose.
2. Derive an analytical expression for the line which passes through a point with coordinates
(a1, a2, a3) and which is perpendicular to vectors b = (b1, b2, b3)" and c = (c1, c2, C3)".
%3D
1.2.4 Prove that vectors (1, 1, 1)", (2, 1,0)", (3, 1, 4)", and (1,2, –2)" are linearly dependent.
1.2.5 If a, b, and c are linearly independent vectors in R", prove that a + b, b+c, and a+c
are also linearly independent. Is the same true of a – b, b+ c, and a + c?
1.3.3 Give an example where rank(AB) + rank(BA). (Hint: Try some 2 x 2 matrices.)
Transcribed Image Text:1. Let A and B be n xn matrices such that det A ±0 and det B 70. Prove that ((AB)")-1 = (A-1)"(B-1)". Hint: Use the properties of the inverse and transpose. 2. Derive an analytical expression for the line which passes through a point with coordinates (a1, a2, a3) and which is perpendicular to vectors b = (b1, b2, b3)" and c = (c1, c2, C3)". %3D 1.2.4 Prove that vectors (1, 1, 1)", (2, 1,0)", (3, 1, 4)", and (1,2, –2)" are linearly dependent. 1.2.5 If a, b, and c are linearly independent vectors in R", prove that a + b, b+c, and a+c are also linearly independent. Is the same true of a – b, b+ c, and a + c? 1.3.3 Give an example where rank(AB) + rank(BA). (Hint: Try some 2 x 2 matrices.)
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