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 real matrices. Assume that
is invertible. Show that
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 is also invertible.
Observe that , taking determinant is both sides as A+B , A and B are invertible so |A+B| , |A| and |B| are non-zero . Hence
is also non-singular .
Again we have , , taking inverse on both sides .
Now as A+B , A and B are invertible so , we have . Hence we are done .
If are non-singular matrices of the same order such that
and
are also non-singular, then find the value of
.
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 real matrices. Assume that
is invertible. Show that
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 is also invertible.
Observe that , taking determinant is both sides as A+B , A and B are invertible so |A+B| , |A| and |B| are non-zero . Hence
is also non-singular .
Again we have , , taking inverse on both sides .
Now as A+B , A and B are invertible so , we have . Hence we are done .
If are non-singular matrices of the same order such that
and
are also non-singular, then find the value of
.