In parts (a)-(l) determine whether the statement is true or false, and justify y
ID: 3122946 • Letter: I
Question
In parts (a)-(l) determine whether the statement is true or false, and justify your answer. If A is a 3 times 3 matrix, then det(2A) = 2 det(A) If A and B are square matrices of the same size such that det(A) = det(B), then det(A + B) = 2 det(A). If A and B are square matrices of the same size and A is invertible, then det(A^- 1 BA) det(B) A square matrix A is invertible if and only if det(A) = 0. The matrix of cofactors of A is precisely [adj(A)]^T. For every n times n matrix A, we have A middot adj(A) = (det(A))I_n If A is a square matrix and the linear system Ax = 0 has multiple solutions for x, then det(A) = 0. 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. If E is an elementary matrix, then Ex= 0 has only trivial solution. If A is an invertible matrix, then the linear system Ax = 0 has only thr trivial solution if and only if the linear system A^- 1x = 0 has only the trivial solution. If A is invertible, then adj(A) must also be invertible. If A has a row of zeros, then so does adj(A).Explanation / Answer
1) true as A is a square matrix and 2 is scalar.
2)false take A=[1,1;1,2] =B
3) true as A and B are square matrices so det(A^-1 BA) = det(A^-1)det(B) det A
4)false invertible if determinant is non zero
5) true
6) true
7) false
8) true
9) true
10) true
11) true
12) false adj A has column of zeroes