Construct 3 ×2 matrices A and B such that A 𝐱=0 has only the trivial solution and B 𝐱=0 has a no

step 1 In this example, we're going to let ABA3 by two matrix. Then let's consider the matrix equation,

Construct 3 ×2 matrices A and B such that A 𝐱=0 has only the...

Key Concepts

Matrix Rank

The rank of a matrix is the maximum number of linearly independent columns (or rows) in the matrix. A matrix with full column rank (rank equal to the number of columns) will have only the trivial solution in its homogeneous system. If the rank is less than the number of columns, then there exist free variables, resulting in a nontrivial nullspace and thus nontrivial solutions.

Linear Independence

Linear independence of vectors (or the columns of a matrix) means that no vector in the set can be written as a linear combination of the others. For a homogeneous system Ax = 0 to have only the trivial solution, the columns of A must be linearly independent. If the columns are linearly dependent, then nonzero vectors exist in the nullspace, leading to nontrivial solutions.

Homogeneous Systems

A homogeneous system of linear equations is one in which the constant terms are zero, so the system takes the form Ax = 0. This system always has at least the trivial solution (x = 0), but depending on the matrix A, additional nontrivial solutions may exist. The structure of the solution set is given by the nullspace of the matrix.

Nullspace (Kernel)

The nullspace of a matrix A consists of all vectors x that satisfy Ax = 0. The properties of the nullspace—such as whether it contains only the zero vector (trivial nullspace) or other vectors (nontrivial nullspace)—depend on the linear independence of the columns of the matrix. In the context of the problem, one matrix is chosen so its nullspace is trivial, while the other matrix has a nontrivial nullspace.

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