Question

If the matrix A is of order 3x3 and the system of equations AX = B has a unique solution, what can be concluded about the determinant of A?

• A. The determinant of A is zero
• B. The determinant of A is non-zero
• C. The determinant of A must be 1 only
• D. The determinant of A cannot be negative

Hint:

For a system of linear equations \(AX=B\):

• If \(\det(A) \neq 0\), there is a unique solution.

• If \(\det(A) = 0\) and \((\text{adj } A)B \neq 0\), there is no solution (inconsistent system).

• If \(\det(A) = 0\) and \((\text{adj } A)B = 0\), there are infinitely many solutions (consistent system).
This summary is crucial for solving problems related to the nature of solutions of linear equations.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are dealing with a system of linear equations represented in matrix form as \(AX = B\), where A is a 3x3 coefficient matrix.
The question states that this system has a unique solution and asks about the property of the determinant of matrix A.

Step 2: Key Formula or Approach:
According to the Cramer's rule and matrix inversion method for solving systems of linear equations, a system \(AX = B\) has a unique solution if and only if the coefficient matrix A is invertible (non-singular).
A matrix is invertible if and only if its determinant is non-zero.

Step 3: Detailed Explanation:
The system of linear equations is given by \(AX = B\).
If the matrix A is invertible, we can find its inverse, \(A^{-1}\).
Multiplying the equation by \(A^{-1}\) on the left, we get:
\[ A^{-1}(AX) = A^{-1}B \]
\[ (A^{-1}A)X = A^{-1}B \]
\[ IX = A^{-1}B \]
\[ X = A^{-1}B \]
This equation gives a unique solution for the variable matrix X.
The condition for the existence of \(A^{-1}\) is that the determinant of A must be non-zero (\(\det(A) \neq 0\)).
If \(\det(A) = 0\), the matrix A is singular, and its inverse does not exist. In this case, the system of equations will have either no solution or infinitely many solutions, but not a unique solution.
Since the problem states that the system has a unique solution, it is necessary that the determinant of A is non-zero.
The value can be any non-zero real number; it is not restricted to be 1 or only positive.

Step 4: Final Answer:
For the system of equations \(AX = B\) to have a unique solution, the determinant of the coefficient matrix A must be non-zero.