Final Answer:
The inverse of the matrix A=[-3 1; 4 3] does not exist due to its singularity. Matrix inversion is only possible for non-singular matrices, and in this case, the row operations lead to inconsistency or a row of zeros.
Step-by-step explanation:
The existence of the inverse of a matrix is determined by whether the matrix is invertible or singular. In this case, to find the inverse of the matrix A=[-3 1; 4 3], one can use row operations to transform it to its reduced row-echelon form. If the resulting matrix has a row of zeros or is not in the form of the identity matrix, then the original matrix is singular, indicating that its inverse does not exist.
In this specific scenario, the row operations may lead to an inconsistent system or a row of zeros, making it impossible to obtain the identity matrix. Therefore, the inverse of the given matrix is not attainable.
Matrix inversion involves various methods, such as Gaussian elimination and Cramer's rule. Understanding the conditions for the existence of an inverse and recognizing singular matrices is crucial in linear algebra and mathematical applications.