For each of the following statements decide whether it is true (T) or false (F).

Question

For each of the following statements decide whether it is true (T) or false (F). Substantiate the statement your decision (a reminder: in case you wish to prove that the statement is false, a single counterexample provides a sufficient argument). (a) For two invertible n times n matrices A and B, det (AB) = 1/det A det B. (b) det (2A) = 2det A (c) 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 (P^2AP^-1) = det A det P. (e) The determinant of the RREF of an invertible matrix can be equal to 0. (f) A system of linear equations is Cramer's iff the associated homogeneous system has only the trivial solution.

Explanation / Answer

c) if one of the column is multiplied by -1 thetn detA becomes -detA

If rows are interchanged then -detA remains the same

So, False

d) det(P^2AP^-1) = det(P^2)(detA)(1/detP)

= det(P)^2(detA)(1/detP)

= detP*(detA)

True

e) determinat of Rrefr of an ivertible matrix cannot be zero

False

f) Homogeneous system of linear equations gives solution as x1,x2,x3.....= 0

As All the determinants D1, D2, …, Dn however will be zero, since we are substituting an entire column filled with zero into each of them

True

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