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In system of Linear equations Ax=B, where A is m*n ,which of the following is/are true?

a:) If b=0, and m=n,then only trivial solution is possible.

b:) If b=0, and m=n,then always trivial solution is possible.

c:)If b=0,and m<n ,then always non-trivial solution possible.

c:)If b=0,and m<n ,then only non-trivial solution possible.

d:)If b=0,and m>n ,then only non-trivial solution possible.

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a) is false. Only trivial solution is possible only if Rank[A] = n

b) is true. For homogenous system of equations i.e. of the type AX=0, X = 0 is always a solution also known as the trivial solution

c) is true. m $\leq$ n implies Rank[A] $\leq$ m. So the system must possess an infinite number of non trivial solutions.

d) is not true as X=0 is always a solution which is the trivial solution. Hence saying that 'only' non-trivial solutions exist is not true.

e) For the last case refer this https://en.wikipedia.org/wiki/Overdetermined_system

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