Which of the following is NOT equivalent to the nxn matrix A being invertible? A. The homogeneous system associated to A has a unique solution. B. Some non-homogeneous system whose coefficient matrix is A has a unique solution. C. Every non-homogeneous system whose coefficient matrix is A is consistent. D. The column space of A is R". E. The linear transformation x H Ax is one-to-one.

For the matrix A of order [tex]n\times n[/tex] to be invertible, its determinant must not be equal to zero, |A| [tex]\neq[/tex]0, [tex]A^{- 1}[/tex] exists if- AC = CA = I, where I is identity matrix.

AB = 0, B = [tex]A^{- 1}.0 = 0[/tex]

Thus, B = (0, 0, 0......., 0) is a unique solution

2. The non - homogeneous equation system with coefficient matrix A has a unique solution:

For an equation- AY = D

Y = [tex]A^{- 1}.D[/tex] is a unique solution

3. Every non homogeneous equation with coefficient matrix A is not consistent as:

For an equation- AY = D, has a solution.l Thus coefficient matrix is inconsistent whereas augmented matrix is.

4. Rank of matrix A = n, Thus the column space of A is [tex]R^{n}[/tex]

5. Since, column space of A = [tex]R^{n}[/tex], thus x→xA is one-to-one