Question

Suppose A is an invertible matrix and the rows of A each sum to one. What can you say about the rows of A −1?

Hint: phrase this as a transformation problem.

Answer 1

When a matrix A is invertible and the rows of A each sum to one, the rows of A^(-1) also sum to one.

Step-by-step explanation:

When a matrix A is invertible and the rows of A each sum to one, we can say that the rows of A^(-1) also sum to one. This can be understood as a transformation problem. Let's consider a vector v whose components are the sums of the rows of A. Now, when we multiply A^(-1) by v, we get the vector whose components are the sums of the rows of A^(-1). Since the rows of A^(-1) represent the coefficients of a new system of equations, the sums of these rows will also be equal to one.