Matrix Properties Exploring Rank And Inverses Of A And B Where AB Equals I4

In the realm of linear algebra, the study of matrices and their properties forms a cornerstone for numerous applications across various scientific and engineering disciplines. When dealing with matrix multiplication, the resulting matrix's rank and the individual matrices' characteristics provide valuable insights into the underlying linear transformations and system behaviors. This article delves into a specific scenario involving two matrices, A and B, with dimensions 4 × 7 and 7 × 4 respectively, where their product AB results in the 4 × 4 identity matrix, denoted as I₄. We aim to dissect the implications of this relationship, focusing on the ranks of A and B, and whether A possesses a right inverse.

We are given a 4 × 7 real matrix A and a 7 × 4 matrix B, such that their product, AB, equals the 4 × 4 identity matrix I₄. The fundamental question lies in determining the properties of A and B based on this relationship. Specifically, we need to ascertain the ranks of both matrices and whether A has a right inverse. The identity matrix I₄ is a square matrix with ones on the main diagonal and zeros elsewhere, representing a transformation that leaves vectors unchanged. The fact that AB equals I₄ implies that the transformation represented by AB effectively acts as an identity transformation in a 4-dimensional space. This has significant consequences for the individual transformations represented by A and B.

Determining the rank of matrix A is crucial. Recall that the rank of a matrix represents the maximum number of linearly independent rows or columns it contains. Given that A is a 4 × 7 matrix, its rank can be at most 4, the smaller of its dimensions. The key here is the equation AB = I₄. A fundamental property of matrix ranks states that for any two matrices A and B that can be multiplied, the rank of their product, rank(AB), is less than or equal to the minimum of the ranks of the individual matrices: rank(AB) ≤ min(rank(A), rank(B)). In our case, rank(I₄) = 4. Therefore, we have 4 ≤ min(rank(A), rank(B)). This inequality immediately tells us that rank(A) must be at least 4. Since the maximum possible rank for A is 4, we conclude that rank(A) = 4. This means that the rows of A are linearly independent, and A spans the entire 4-dimensional space. This result is vital for understanding the properties of A and its role in the transformation process.

Now let's analyze the rank of matrix B. B is a 7 × 4 matrix, so its rank can be at most 4, which is the smaller of its dimensions. From the inequality 4 ≤ min(rank(A), rank(B)), we know that rank(B) must be at least 4. Therefore, rank(B) is also equal to 4. This indicates that the columns of B span a 4-dimensional subspace within the 7-dimensional space. If rank(B) were 7, it would imply that B is invertible, which is not possible since B is not a square matrix. The rank of B being 4 implies that there are 4 linearly independent columns in B, which is crucial for the transformation from a 4-dimensional space back to a subspace within the 7-dimensional space after A transforms a 7-dimensional space to a 4-dimensional space.

The relationship AB = I₄ has another significant implication: B serves as a right inverse of A. A right inverse of a matrix A is a matrix B such that AB = I, where I is the identity matrix. The existence of a right inverse for A is directly linked to its rank. Since we have already established that rank(A) = 4, which is the maximum possible rank for a 4 × 7 matrix, A has a right inverse. This means that the transformation represented by A maps the 7-dimensional space onto the entire 4-dimensional space. The right inverse B then maps the 4-dimensional space back into a 4-dimensional subspace within the 7-dimensional space, effectively undoing the transformation of A within that subspace. However, A does not have a left inverse because if it did, that would imply that A is a square matrix, which contradicts the given dimensions of A.

The fact that AB = I₄ tells us a lot about the transformations represented by A and B. Matrix A maps a 7-dimensional space onto a 4-dimensional space, and matrix B provides a way to map back from the 4-dimensional space to a subspace within the original 7-dimensional space. The rank of A being 4 signifies that A's rows are linearly independent, ensuring that the transformation covers the entire 4-dimensional space. The rank of B also being 4 indicates that the transformation back into the 7-dimensional space is not a full reversal but rather a mapping into a 4-dimensional subspace. This understanding is critical in various applications such as solving systems of linear equations, data compression, and signal processing, where transformations between different dimensional spaces are common. The existence of the right inverse B for A is a key element in ensuring that there is a way to at least partially undo the transformation performed by A, which is crucial for maintaining information and solving inverse problems.

In summary, given a 4 × 7 matrix A and a 7 × 4 matrix B such that AB = I₄, we have determined that the rank(A) = 4 and the rank(B) = 4. Furthermore, A has a right inverse, which is B. These findings are grounded in the fundamental principles of linear algebra, particularly the properties of matrix ranks and inverses. Understanding these relationships provides a deeper insight into the transformations represented by these matrices and their applications in various fields. The interplay between the dimensions of the matrices, their ranks, and the resulting identity matrix unveils the intricate dance of linear transformations between different dimensional spaces. This underscores the importance of these concepts in solving complex problems across science and engineering, where matrix algebra provides a powerful toolkit for modeling and analyzing systems.