Show that if a matrix A ∈ℂ^p ×q has two right inverses B1 and B2, then λB1+(1-λ) B2 is also a ri

Step 1: Understand the definition of a right inverse. A matrix $B$ is a right inverse of a matri

Show that if a matrix A ∈ℂ^p ×q has two right inverses B1...

Show that if a matrix $A \in \mathbb{C}^{p \times q}$ has two right inverses $B_1$ and $B_2$, then $\lambda B_1+(1-\lambda) B_2$ is also a right inverse for every choice of $\lambda \in \mathbb{C}$.

Key Concepts

Right Inverse

A right inverse of a matrix A is a matrix B such that the product AB equals the appropriate identity matrix. This concept is fundamental in linear algebra as it indicates that even if A is not square or invertible in the usual sense, there may exist a matrix B that acts as a partial inverse, at least from the right. It is used to solve systems of linear equations where the solution can be represented in terms of a right inverse of the coefficient matrix.

Distributive Property of Matrix Multiplication

The distributive property of matrix multiplication over addition allows one to combine and simplify expressions such as A(?B? + (1-?)B?) into ?(AB?) + (1-?)(AB?). This property is essential in demonstrating that if each term AB? and AB? equals the identity matrix, then any linear combination of B? and B? still preserves the right inverse property when multiplied by A.

Linear Combination of Right Inverses

When a matrix A has more than one right inverse, any linear combination of these inverses—when scaled and summed appropriately—remains a right inverse for A, as long as the scalar coefficients sum to one. This is because A applied to such a combination yields the same linear combination of identity matrices, which simplifies back to the identity. This concept illustrates how solutions to an underdetermined linear system can form an affine space.

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