Linear Algebra Problem Please explain in detail Perhaps it

[ Linear Algebra Problem ]

Please explain in detail. Perhaps it has something to do with the ranks?

Solution

(24)

we are given A as 4X4 matrix

b and c are two vectors in R^4

Ax=b has unique solution

It means that b has unique solution

then we will get rref(A) will have a leading one in each column

now, we know that A has same number of rows and column

then there will be no rows in which all values are zero and there will also not be any free variables

Hence , Ax=c will have unique solution........Answer

(25)

we are given A as 4X4 matrix

b and c are two vectors in R^4

Ax=b has inconsistent

It means that Ax=b will have no solution

or b= will have no solution

And it can only possible if det(A)=0

now, if Ax=c

if det(A)=0

then Ax=c will also have no solution........Answer

(26)

we are given A as 4X3 matrix

b and c are two vectors in R^4

Ax=b has unique solution

firstly , we find rank of A

we know formula that

if A is mXn matrix

then rank(A)=min(m,n)

Here m we have A =4X3

rank(A)=min(4,3)

rank(A)=3

and b has unique solution

so, \'c\' will have either no solution or unique solution

But it can not have many solution