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Unformatted text preview: MATH 152 Applied Linear Algebra and Differential Equations Spring 2004-05 2 Review Notes for Linear Algebra – True or False Last Updated: May 10, 2005 The following answers are not guaranteed to be correct. Chapter 5 [ Matrix ] 5.1 If A is a square matrix and A 2 = I , then A = I or A =- I . False . A = • 1 1 ‚ . 5.2 If AB = O , then A = O or B = O . False . A = B = • 1 ‚ . 5.3 For square matrices A , B , C , if ABC = O , then one of them is O . False . A = B = C = • 1 ‚ . 5.4 If AB = AC , then B = C . False . Choose B 6 = C and A = O . 5.5 The square of a nonzero square matrix must be a nonzero matrix. False . • 1 ‚ 2 = • ‚ . 5.6 If AB = BA , then ( A + B ) 3 = A 3 + 3 A 2 B + 3 AB 2 + B 3 . True . ( A + B ) 3 = ( A + B )( A 2 + AB + BA + B 2 ) = A 3 + A 2 B + ABA + AB 2 + BA 2 + BAB + B 2 A + B 3 = A 3 + A 2 B + A 2 B + AB 2 + A 2 B + AB 2 + AB 2 + B 3 = A 3 + 3 A 2 B + 3 AB 2 + B 3 . 5.7 An invertible matrix must be a square matrix. True . A is invertible ⇐⇒ AB = BA = I for some B . If A is m × n and B is n × m , then we must have m = n . 5.8 A non-square matrix can never be invertible. True . Equivalent statement as 5.7 . 5.9 If A has a zero row or a zero column, then A is not invertible. True . A has zero determinant and hence not invertible. 5.10 Let A , B be invertible matrices. Then A + B is also invertible. False . Let A invertible. Choose B =- A . Then B invertible but A + B = O not invertible. 5.11 If AB is equal to the identity matrix, then A must be an invertible matrix. False . Choose A , B non-square like A = • 1 1 ‚ , B = 1 1 . 1 5.12 A , B are square matrices. If AB = I , then BA = I . Hence, A is invertible. True . We prove it by contradiction. In the following proof, we need so-called elementary matrices E i , for instance, like 1 1 1 , 1- 6 1 , 1 1- 4 1 . In this case, these three matrices correspond to the elementary row operations R 2 ↔ R 3 ,- 6 R 2 ,- 4 R 1 + R 3 , respectively. In general, each row operation always has a corresponding elementary matrix which is also invertible. So, if A is row equivalent to B by doing some row operations to A , then we also mean B = E s ··· E 2 E 1 A for some invertible E 1 , E 2 , ··· , E s . Proof: Suppose the square matrix A is not invertible. Then A is not row equivalent to I = ⇒ the row echelon form of A must have a zero row = ⇒ E s ··· E 2 E 1 A has a zero row = ⇒ E s ··· E 2 E 1 AB = E s ··· E 2 E 1 = invertible matrix also has a zero row = ⇒ a contradiction. This proves A must be invertible. 5.13 For square matrix A , AA t = I if and only if A t A = I . True . If A , B are square matrices, then AB = I ⇐⇒ BA = I ....
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