Question

Consider the system of linear equations \(Ax=b\), where \(A\) is an \(n\times n\) 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?

• A. \(A^{-1}\) exists
• B. The system of equations \(A^mx=b\) also has a unique solution for \(m=1,2,3,\ldots\)
• C. \(\text{rank}(A)=\text{rank}(A^m)\), for \(m=1,2,3,\ldots\)
• D. \(\text{rank}(A)<\text{rank}([A\,|\,b])\), where \([A\,|\,b]\) denotes the augmented matrix

Hint:

A unique solution to a square system means A is invertible; use that to check invertibility of powers of A and the rank of the augmented matrix.

Answer 1

The Correct Option is A, B, C

Step 1: Translate the phrase unique solution into a statement about $\det(A)$.
For a square system, a unique solution exists exactly when $\det(A)\neq0$. So we are told $\det(A)\neq0$.

Step 2: Settle statement (A) directly.
A matrix has an inverse exactly when its determinant is nonzero. Since $\det(A)\neq0$, $A^{-1}$ exists. Statement (A) is correct.

Step 3: Settle statement (B) using determinants of powers.
\[ \det(A^m)=(\det A)^m \]
Since $\det(A)\neq0$, raising it to any positive integer power $m$ keeps it nonzero, so $\det(A^m)\neq0$ for every $m=1,2,3,\ldots$. This means $A^m$ is invertible, so $A^mx=b$ has a unique solution for every $m$. Statement (B) is correct.

Step 4: Settle statement (C) using the same determinant fact.
Since $\det(A^m)\neq0$ for every $m$, $A^m$ is a full rank $n\times n$ matrix for every $m$, meaning $\text{rank}(A^m)=n$. Also $\text{rank}(A)=n$ because $\det(A)\neq0$. So $\text{rank}(A)=\text{rank}(A^m)=n$ for every $m$. Statement (C) is correct.

Step 5: Settle statement (D) using a row count argument.
The augmented matrix $[A\,|\,b]$ only has $n$ rows, no matter how many columns it has, so its rank cannot exceed $n$. We already know $\text{rank}(A)=n$, which is the maximum possible value. So $\text{rank}([A\,|\,b])$ cannot be strictly greater than $\text{rank}(A)$; both equal $n$. Statement (D), which claims a strict inequality, is false.

Step 6: Conclude.
\[ \boxed{\text{(A), (B), and (C) are correct}} \]