Question

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.

Answer 1

Answer:

Option C

Step-by-step explanation:

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

1. The homogeneous equation with coefficient matrix A has a unique solution:

       AB = 0, B = $A^{- 1}.0 = 0$

      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 = $A^{- 1}.D$ 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 $R^{n}$

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