Consider the system of linear equations: Ax = b, where A is an n×n matrix, and x and b are n-dimensional column vectors. Suppose this system of equations has a unique solution. Which of the following statements is/are correct?

The correct statements are:

• A⁻¹ exists
• The system A^mx = b also has a unique solution for m = 1,2,3,...
• rank(A) = rank(A^m), for m = 1,2,3,...

Step-by-Step Explanation:

We are given a system of n linear equations with n variables:

$$Ax=b$$

where A is an n×n square matrix, and x and b are n-dimensional column vectors. We are given that this system has a unique solution.

1. Analysis of the statement: A⁻¹ exists
For a square matrix A, the system of equations Ax = b has a unique solution if and only if the matrix A is non-singular (invertible). If a matrix is invertible, its inverse matrix exists. Thus, the matrix A⁻¹ exists.
This statement is correct.

2. Analysis of the statement: The system A^mx = b also has a unique solution for m = 1,2,3,...
Since A is invertible, we know that the determinant of A is non-zero:

$$det\left(A\right)\ne 0$$

Using the properties of determinants, the determinant of A^m for any positive integer m is:

$$det\left(A^m\right)={[det(A\left)\right]}^m$$

Since det(A) ≠ 0, it follows that det(A^m) ≠ 0 for all m = 1,2,3,... This means that the matrix A^m is also invertible (its inverse is (A⁻¹)^m). Because A^m is invertible, the system of equations A^mx = b has a unique solution given by:

$$x={\left(A^m\right)}^{-1}b$$

This statement is correct.

3. Analysis of the statement: rank(A) = rank(A^m), for m = 1,2,3,...
For any invertible n×n matrix, its rank is equal to n (full rank). Since A is invertible, we have:

$$\text{rank}\left(A\right)=n$$

As established above, A^m is also an invertible n×n matrix for any positive integer m. Therefore, its rank is also full:

$$\text{rank}\left(A^m\right)=n$$

Comparing the two ranks, we get rank(A) = rank(A^m) = n for all m = 1,2,3,...
This statement is correct.

4. Analysis of the statement: rank(A) < rank([A|b]), where [A|b] denotes the augmented matrix
According to the Rouché-Capelli theorem, a system of linear equations Ax = b is consistent (possesses at least one solution) if and only if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A|b]. Since we are given that a unique solution exists, the system must be consistent, which implies:

$$\text{rank}\left(A\right)=\text{rank}\left(\right[A\left|b\right])$$

Therefore, the inequality rank(A) < rank([A|b]) is false.
This statement is incorrect.