I need urgent help on this, thanks... Indicate wheter the statament is always tr

Question

I need urgent help on this, thanks...
Indicate wheter the statament is always true or sometimesfalse. Justify your answer by giving a logical argument or acounterexample. a) If A is a singular n x n matrix, then Ax =0 has indefinitely many solutions. b) If A is a singular n x n matrix, then the reducedrow-echelon form of A has at least one row of zeros. c) If A-1 is expressible as aproduct of elementary matrices, then the homogenous linear systemAx = 0 has only the trivial solution. d) If A is singular n x n matrix, andB results ny interchanging two rows of A, thenB may or may not be singular. I need urgent help on this, thanks...
Indicate wheter the statament is always true or sometimesfalse. Justify your answer by giving a logical argument or acounterexample. a) If A is a singular n x n matrix, then Ax =0 has indefinitely many solutions. b) If A is a singular n x n matrix, then the reducedrow-echelon form of A has at least one row of zeros. c) If A-1 is expressible as aproduct of elementary matrices, then the homogenous linear systemAx = 0 has only the trivial solution. d) If A is singular n x n matrix, andB results ny interchanging two rows of A, thenB may or may not be singular.

Explanation / Answer

a) TRUE the determinant of a nonsingular matrix A of order nxn isnonzero implies rank of A is n and so all the rows in the reducedechelon form of A are nonzero implies Ax = 0 is with only trivial solution b) TRUE if A is singular then the rank of A is less than (<)n implies there exist a zero row in the reduced echelon form ofA c) TRUE d) FALSE given that A is singular implies det A = 0 B is the matrix obtained by interchanging two rows of A note that the det of A is invariant by interchanging any tworows of A thus det B = 0 implies B is singular

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