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1. Consider the determinant as a sum of signed products. Recall that there are n! elementary products in the determinant expansion of an arbitrary n x n matrix A. (a) If A is a 5 x 5 matrix, how many of the elementary products in det(A) are guaranteed to be zero if the entry a12 is zero? (b) If A is an nxn matrix, n ≥ 2, how many of the elementary products in det (4) are guaranteed to be zero if a12 = 0? Give your answer in terms of n. (c) If A is an nxn matrix with det (A) #0, what is the maximum possible number of zero entries in A? Give your answer in terms of n.

1. Consider the determinant as a sum of signed products. Recall that there are n! elementary products in the determinant expansion of an arbitrary n x n matrix A. (a) If A is a 5 x 5 matrix, how many of the elementary products in det(A) are guaranteed to be zero if the entry a12 is zero? (b) If A is an nxn matrix, n ≥ 2, how many of the elementary products in det (4) are guaranteed to be zero if a12 = 0? Give your answer in terms of n. (c) If A is an nxn matrix with det (A) #0, what is the maximum possible number of zero entries in A? Give your answer in terms of n.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Consider the determinant as a sum of signed products. Recall that there are n! elementary products in the determinant expansion of an arbitrary n x n matrix A. (a) If A is a 5 x 5 matrix, how many of the elementary products in det(A) are guaranteed to be zero if the entry a12 is zero? (b) If A is an nxn matrix, n ≥ 2, how many of the elementary products in det (A) are guaranteed to be zero if a12 = 0? Give your answer in terms of n. (c) If A is an nxn matrix with det (A) #0, what is the maximum possible number of zero entries in A? Give your answer in terms of n.
Transcribed Image Text:4. Consider the determinant as a sum of signed products. Recall that there are n! elementary products in the determinant expansion of an arbitrary n x n matrix A. (a) If A is a 5 x 5 matrix, how many of the elementary products in det(A) are guaranteed to be zero if the entry a12 is zero? (b) If A is an nxn matrix, n ≥ 2, how many of the elementary products in det (A) are guaranteed to be zero if a12 = 0? Give your answer in terms of n. (c) If A is an nxn matrix with det (A) #0, what is the maximum possible number of zero entries in A? Give your answer in terms of n.
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