Question

Let \(A\) be \(n \times n, n \geq 2,\) and assume one column of \(A\) consists of zeros. Find the possible values of \(\operatorname{rank}(\operatorname{adj} A)\)

Short answer

The rank of \(\operatorname{adj}(A)\) is either 0 or 1.

Step-by-step solution

Understanding the Rank of the Adjugate Matrix

Given that matrix \(A\) is an \(n \times n\) matrix and one of its columns consists entirely of zeros, observe that this directly implies that \(A\) is a singular matrix (it has no full-rank), hence \(\det(A) = 0\). Therefore, any calculation related to \(A\)'s adjugate or adjoint should take this into account.

Properties of the Adjugate Matrix

The adjugate matrix, \(\operatorname{adj}(A)\), is the transpose of the cofactor matrix of \(A\). A key property states that \(A \cdot \operatorname{adj}(A) = \det(A) \cdot I\), where \(I\) is the identity matrix. Since \(\det(A) = 0\), it follows that \(A \cdot \operatorname{adj}(A) = 0\cdot I = 0\). This indicates that each column of \(\operatorname{adj}(A)\) is in the null space of \(A\).

Possible Values for \(\operatorname{rank}(\operatorname{adj} A)\)

Since \(A\) is singular and \(A \cdot \operatorname{adj}(A) = 0\), the adjugate matrix \(\operatorname{adj}(A)\) will be either the zero matrix or will have a rank of 1, depending on the null space dimension. If the null space contains only the zero vector, then \(\operatorname{adj}(A)\) must itself be the zero matrix (leading to a rank of 0). Generally, the rank of \(\operatorname{adj}(A)\) could also be 1, depending on if there are non-zero vectors in the null space.

Conclusion

Therefore, the possible values for the rank of \(\operatorname{adj}(A)\) when \(A\) is a square matrix with a zero column, is \(0\) or \(1\). In general, the rank of \(\operatorname{adj}(A)\) for a singular matrix can be \(0\) through \(n-1\), but a column of zeros constrains this to \(0\) or \(1\).

Key concepts

Singular Matrix

A singular matrix is a square matrix that does not have an inverse. Understanding this concept fully helps us grasp why certain operations involving matrices may not be possible. A key characteristic of a singular matrix is that its determinant is zero, indicated as \( \det(A) = 0 \). This implies that the matrix lacks full rank, meaning it does not reach the maximum number of linearly independent rows or columns. In our case, if a column of the matrix consists entirely of zeros, the matrix instantly becomes singular. This zero column indicates that the matrix cannot span the entire space, which restricts certain computations and transformations. When dealing with singular matrices, these properties will affect any further calculations, especially when considering the adjugate or the null space of the matrix.

Cofactor Matrix

The cofactor matrix of a given square matrix \( A \) is derived by calculating the cofactor for each element within \( A \). A cofactor is essentially the determinant of a smaller matrix that results when a particular row and column are removed from \( A \). These cofactors are then arranged in a collectively structured matrix that is directly related to the adjugate (or adjoint) matrix. The adjugate matrix \( \operatorname{adj}(A) \), which is the transpose of the cofactor matrix, plays an integral role in matrix equations like \( A \cdot \operatorname{adj}(A) = \det(A) \cdot I \). Here, the identity matrix \( I \) is a diagonal matrix with ones on the diagonal. This equation becomes crucial when \( \det(A) = 0 \), as it simplifies to \( A \cdot \operatorname{adj}(A) = 0 \). It tells us that every linear combination of the columns in the adjugate matrix results in a zero vector, emphasizing that they lie within the null space of \( A \). The intricate link between these matrices underlines their interdependence when evaluating matrix properties, especially in scenarios involving singular matrices.

Null Space

The null space of a matrix \( A \) is a set of all vectors that, when \( A \) is multiplied by a vector, results in the zero vector. Mathematically, it's defined as all solutions to the equation \( A\mathbf{x} = \mathbf{0} \). Understanding the null space is important because it reflects the solutions to homogeneous systems of linear equations derived from \( A \). When the adjugate matrix \( \operatorname{adj}(A) \) is computed for a singular matrix \( A \), each column of this matrix will be in the null space of \( A \), emphasizing their interrelation.If \( A \) has a column of zeros, then the rank and nullity theorem, which connects the rank of a matrix to the dimension of its null space, plays a significant role. Here, having a non-zero null space infers a presence of non-trivial solutions, which ultimately determines whether \( \operatorname{adj}(A) \) may have non-zero entries. This is why when \( A \) is singular, the null space analysis helps anticipate the possible rank values for \( \operatorname{adj}(A) \), specifically leading us to conclude that the rank could either be \( 0 \) or \( 1 \) when a column is entirely zero.