True-False Exercises In parts (a)-(1) determine whether the statement is true or

Question

True-False Exercises In parts (a)-(1) determine whether the statement is true or false, and justify your answer. (a) If A is a 3 times 3 matrix, then det(2A) = 2 det(A). (b) If A and B are square matrices of the same size such that det(A) = det(B), then det(A + B) = 2 det(A). (b) If A and B are square matrices of the same size such that det(A) = det(B), then det(A + B) = 2 det(A). (c) If A and B are square matrices of the same size and A is invertible, then det(A^-1 BA) = det(B) (d) A square matrix A is invertible if and only if det(A) = 0. (e) The matrix of cofactors of A is precisely [adj(A)]^T. (f) For every n times n matrix A, we have A middot adj(A) = (det(A)) I_n (g) If A is a square matrix and the linear system Ax = 0 has multiple solutions for x, then det(A) = 0. (h) If A is an n times n matrix and there exists an n times 1 matrix b such that the linear system Ax = b has no solutions, then the reduced row echelon form of A cannot be I_n. (i) If E is an elementary matrix, then Ex = 0 has only the trivial solution. (j) If A is an invertible matrix, then the linear system Ax = 0 has only the trivial solution if and only if the linear system A^-1 x = 0 has only the trivial solution. (k) If A is invertible, then adj(A) must also be invertible. (l) If A has a row of zeros, then so does adj(A).

Explanation / Answer

(A)false

(B)false

(C)true

(D)false

(E)true

(F)true

(G)false

(H)true

(I)false

(J)true

(K)true

(L)false

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