What is the solution of the system of equations A X=0 if A^-1 exists? Discuss the solution of A

step 1 So we're discussing what our solution of x looks like, given that a inverse exists or doesn't ex

What is the solution of the system of equations A X=0 if...

Key Concepts

Singular Matrix

A singular matrix is one that does not have an inverse, usually because its determinant is zero. In a homogeneous system AX = 0, if A is singular, the equation may have infinitely many solutions since the nontrivial null space implies that there are nonzero vectors that satisfy the system. This concept is essential in linear algebra as it contrasts with the case of an invertible matrix and underscores the dependency of a system's solution on the nature of the coefficient matrix.

Null Space

The null space (or kernel) of a matrix A is the set of all vectors X that satisfy the equation AX = 0. It reflects the degree of freedom in the solution to a homogeneous system. When A is invertible, its null space is trivial (consisting only of the zero vector). However, if A is not invertible, its null space is nontrivial, meaning there exist nonzero vectors X that satisfy AX = 0, typically leading to an infinite number of solutions.

Homogeneous System

A homogeneous system of linear equations is one in which the constant vector is zero (AX = 0). Such systems always have at least the trivial solution (X = 0). The structure of the solution set, called the null space, depends on the properties of the coefficient matrix. When the matrix is invertible, the null space contains only the zero vector, while if the matrix is singular, it contains infinitely many solutions.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is a matrix that possesses an inverse, meaning there exists another matrix which, when multiplied with the original matrix, yields the identity matrix. In the context of a homogeneous system AX = 0, if A is invertible, the only solution is the trivial solution X = 0 because multiplying both sides of the equation by the inverse of A leads directly to X = A?¹0 = 0.

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