Answer each of the following as True or False justifying your answers:

a)  The linear system  has a unique solution.

Solution

For proving it there is a need to take determinant. But for that make it into coefficient form. It will become like this

Now for taking determinant

This means that the determinant of this matrix is not zero. This shows that the linear system is contain a unique solution. It shows that it is true.

Solution

For proving this let consider the two matrix  A and B. the dimension of these matrixes are  

According to the given problem it can be not these matrixes are

Now performing

It will get

Now work for

It will become like this

 Now perform

Work for the left hand side

The answer will be

Its shows that it is a false statement

a)  If  is an  nonsingular matrix, then  is also nonsingular matrix.

For that case, there is need to prove it

Let consider the Matrix A

This will be proved easily by taking determinant

Now in the first step, there subtract row 1 from 3 it will become like this

Now work for the second column. For this case subtract row 4 from row 2. Then after this multiply row 2 by 2 and then subtract row 4 from 2. It will become like this

Now just subtract row 4 from row three it will become like this

Now just multiply the diagonal elements it will become like this

Hence proved left hand side = right hand side

This shows it is a true statement

a)   If  are distinct vectors in  and  is an  matrix such that , then  is invertible.

Solution

For this purpose, there is need to show trivial solution

It will show that

Now just factoring out the A matrix it will become like this

Now the next thing is that just suppose

It shows that A is invertible, that implies only one solution and it is only when   this will give the only solution to the given problem. According to this the equation will become. Now it will become like this

It means A is completely invertible

This statement is also true.

Q−2: Show that if  is a  matrix, then the only solution to   is the  zero matrix.

For proving this, there is need to suppose a matrix of

The matrix  A will be about