What is the solution of the system of equations A X=∅, if A^-1 exists? Discuss the solution of A

step 1 Okay, so here we consider the system of equations, A -X is equal to the empty set. Now, if A -in

What is the solution of the system of equations A X=∅, if...

Key Concepts

Singular Matrix

A singular matrix is a square matrix that does not have an inverse, often because its columns (or rows) are linearly dependent. In a homogeneous system, if the coefficient matrix is singular, the system may have either only the trivial solution or infinitely many solutions, depending on the dimension of the null space. Since the inverse cannot be used in this case, one must analyze the structure of the matrix to determine the nature of the solution set.

Null Space

The null space of a matrix consists of all the vectors that, when multiplied by the matrix, yield the zero vector. In the case of a homogeneous system AX = 0, the set of solutions is exactly the null space of A. When the matrix is singular (not invertible), the null space may contain infinitely many nontrivial vectors, indicating the presence of multiple solutions.

Homogeneous System

A homogeneous system of linear equations is one in which all of the constant terms are zero, typically written in the form AX = 0. Such systems always have at least the trivial solution (where all the variables are zero). The key interest in homogeneous systems is determining whether any nontrivial solutions exist, which depends on the properties of the coefficient matrix.

Invertible Matrix

An invertible matrix is a square matrix that has an inverse, meaning there exists another matrix which, when multiplied with the original matrix, yields the identity matrix. This property allows one to solve linear systems uniquely by multiplying both sides of an equation by the inverse. In the context of a homogeneous system (i.e., with a zero vector on the right-hand side), the existence of an inverse guarantees that the only solution is the trivial one.

Trivial Solution

The trivial solution to a homogeneous system of equations is the solution in which every variable is zero. When the coefficient matrix is invertible, this is the only solution because the inverse can be used to show that no other vector can satisfy the equation.

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