Determine if the statements are true or false 1 If for some

Determine if the statements are true or false.

1. If, for some matrix A and some vectors x and b, we have Ax=b, then b is a linear combination of the column vectors of A.
? True False

2. Let A be an m×n matrix. then the homogeneous system Ax=0 is consistent if and only if the augmented matrix [A|0] has a pivot in every row.

? True False

3. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
? True False

Solution

1. A vector b is a linear combination of the columns of a matrix A is and only if the equation Ax = b has at least one solution.So, it is True.

2. The homogeneous system of linear equations is always consistent. The solution set of a homogeneous system is never empty. In fact, x1 = 0; x2 = 0,..xn = 0 always satisfy such systems. This solution is called the zero or trivial solution. So, False

3.  Elementary row operations on an augmented matrix never change the solution set of the associated linear system. They are used to get reduced matrix and finally get solution.So,True

3. Elemetary row operatio