This is a very beautiful sample problem from ISI MStat PSB 2006 Problem 1 based on Inverse of a matrix. Let's give it a try !!
Let A and B be two invertible \( n \times n\) real matrices. Assume that \( A+B\) is invertible. Show that \( A^{-1}+B^{-1}\) is also invertible.
Matrix Multiplication
Inverse of a matrix
We are given that A,B,A+B are all invertible real matrices . And in this type of problems every information given is a hint to solve the problem let's give a try to use them to show that \( A^{-1}+B^{-1}\) is also invertible.
Observe that ,\( A(A^{-1}+B^{-1})B= (B+A) \Rightarrow |A^{-1}+B^{-1}|=\frac{|A+B|}{|A| |B| } \) taking determinant is both sides as A+B , A and B are invertible so |A+B| , |A| and |B| are non-zero . Hence \(A^{-1}+B^{-1} \) is also non-singular .
Again we have , \( A(A^{-1}+B^{-1})B= (B+A) \Rightarrow B^{-1} {(A^{-1}+B^{-1})}^{-1} A^{-1} = {(A+B)}^{-1} \) , taking inverse on both sides .
Now as A+B , A and B are invertible so , we have \( {(A^{-1}+B^{-1})}^{-1}=B {(A+B)}^{-1} A \) . Hence we are done .
If \( A \& B\) are non-singular matrices of the same order such that \( (A+B) \) and \( \left(A+A B^{-1} A\right) \) are also non-singular, then find the value of \( (A+B)^{-1}+\left(A+A B^{-1} A\right)^{-1} \).
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
