Decide if the given statement is true or false, and give a brief justification f

Question

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. If each element of an n Times n matrix is doubled, then the determinant of the matrix also doubles. True. The determinant of the matrix will double. False. The determinant of the matrix will increase by a factor of n. False. The determinant of the matrix will increase by a factor of 2n. False. The determinant of the matrix will increase by a factor of 2^n. False. The determinant of the matrix will increase by a factor of n^2.

Explanation / Answer

yes, it is true that det of matrix doubles that is

|det(2A)| = 2n|det(A)| for nxn matrix

by geometric argument, if we multiply every element of matrix A by positive number c change expansion of T by the factor c in each of its dimension for n dimension linear transformation this multiplication will change its overall expansion by factor cn

hence, we conclude that for nxn matrix A

|det(cA)| = cn|det(A)|

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