TIFR 2015 Problem 1 Solution -Invertible Matrix with Sum of Each Row 1

TIFR 2015 Problem 1 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programe leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
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Problem:

Let (A) be an invertible (10 \times 10) matrix with real entries such that the sum of each row is 1. Then

A. The sum of the entries of each row of the inverse of A is 1

B. The sum of the entries of each column of the inverse of A is 1

C. The trace of the inverse of A is non-zero.

D. None of the above.

Discussion:

The sum of each row of (A) is 1, means that the sum of the columns of A is the vector ((1,1,...,1)^T ) .

Note that i-th column of (A) is given by (Ae_i ). Therefore, (\sum_{i=1}^{10} Ae_i = (1,1,...,1)^T ).

Since left multiplication by (A) is a linear transformation, the left-hand side of the last expression can be written as (A(\sum_{i=1}^{10}e_i)).

Now, (\sum_{i=1}^{10}e_i = (1,1,...,1)^T ).

Hence we get (A(1,1...,1)^T = (1,1,...,1)^T ).

Another way of saying the last expression is that the vector ( (1,1...,1)^T ) is fixed by A.

Since A is invertible, applying (A^{-1}) on both sides of the last expression we get ((1,...,1)^T = A^{-1}(1,1,...,1)^T ).

By the linearity argument as above, this gives ( (1,1...,1)^T = \sum_{i=1}^{10} A^{-1} (e_i) ). And the i-th term in right-hand side expression is the i-th column of (A^{-1}). Therefore, sum of columns of (A^{-1}) is the vector ((1,1,...,1)^T ). This is same as saying that sum of entries of each row of (A^{-1}) is 1.

Helpdesk

• What is this topic: Linear Algebra
• What are some of the associated concept: Linear Transformation,Invertible Matrix
• Book Suggestions: Introduction to Linear Algebra by Gilbert Strang