For each of the following statements decide whether it is true (T) or false (F).
ID: 3137375 • Letter: F
Question
For each of the following statements decide whether it is true (T) or false (F). Substantiate your decision (a reminder: in case you wish to prove that the statement is false, a single counterexample provides a sufficient argument). 8. (a) For two invertible nxn matrices A and B, det(AB) - det AdetB det Adet B (b) det (2.A)-2det A (c) A determinant does not change as a result of the following two operations: one of its columns is multiplied by -1, and two of its rows are interchanged. (d) If A and P are square matrices of the same size, and P is invertible, det (P2 AP-) det Adet P (e) The determinant of the RREF of an invertible matrix can be equal to 0. (1) A system of linear equations is Cramer's iff the associated homogeneous system has only the trivial solution.Explanation / Answer
8(a). False. det(AB) = det(A).det(B).
(b). False. det(2A) = 2n det(A).
( c). True. If we multiply a column of A by a scalar, the determinant of A will be multiplied by the same scalar. Also, the interchange of 2 rows changes the sign of the determinant.
(d). False. det(P-1 AP) = det(P-1) det(A)det(P) = det(A)det(P)/det(P) = det(A).
(e). False. The interchanging of two rows or columns of a determinant changes the sign of the determinant. Adding a multiple of one row to another does not have any effect on the value of the determinant. Multiplying a row of a determinant by a constant, scales up the value of the determinant by that constant. Only these row operations are used in reducing a matrix to its RREF.
(f). Cannot understand the question. Please elaborate.
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