True-False Review
For items (a)-(j), decide if the given statement is true or false, and give a brief justification for your answer. If true you can quote a relevant definition or theorem from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.
Three
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Differential Equations and Linear Algebra (4th Edition)
- True or False ? a To find the determinant of a triangular matrix, add the entries on the main diagonal. b To find the determinant of a matrix, expand by cofactors in any row or column. c When expanding by cofactors, you need not evaluate the cofactors of zero entries.arrow_forwardGuided Proof Prove that the determinant of an invertible matrix A is equal to 1 when all of the entries of A and A1 is integers. Getting Started: Denote det(A) as x and det(A1) as y. Note that x and y are real numbers. To prove that det(A) is equal to 1, you must show that both x and y are integers such that their product xy is equal to 1. (i) Use the property for the determinant of a matrix product to show that xy=1. (ii) Use the definition of a determinant and the fact that the entries of A and A1 are integers to show that both x=det(A) and y=det(A1) are integers. (iii) Conclude that x=det(A) must be either 1 or 1 because these are the only integer solutions to the equation xy=1arrow_forwardGuided Proof Prove Property 2 of Theorem 3.3: When B is obtained from A by adding a multiple of a row of A to another row of A, detB = detA. Getting Started: To prove that the determinant of B is equal to the determinant of A, you need to show that their respective cofactor expansions are equal. i Begin by letting B be the matrix obtained by adding c times the jth row of A to the ith row of A. ii Find the determinant of B by expanding in this ith row. iii Distribute and then group the terms containing a coefficient of c and those not containing a coefficient of c. iv Show that the sum of the terms not containing a coefficient of c is the determinant of A, and the sum of the terms containing a coefficient of c is equal to 0.arrow_forward
- TEST Only one of the following matrix has an inverse. Find the determinant of each matrix, and use the determinant to identify the one that has an inverse. Then find the inverse. A=[141020101] B=[140020301]arrow_forwardCollinear Points and Determinants a If three points lie on a line, what is the area of the triangle that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points (a1,b1), (a2,b2) and (a3,b3) are collinear if and only if |a1b11a2b21a3b31|=0 b Use a determinant to check whether each set of points is collinear. Graph them to verify your answer. i (6,4), (2,10), (6,13) ii (5,10), (2,6), (15,2)arrow_forward
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