Unizor - Matrices - Inverse Matrix - Problems 1
Channel: Zor Shekhtman on UNIZOR Education
Date: 17/04/2014

1. Prove that an inverse of an identity matrix is the same identity matrix.
In other words, if I is an identity matrix, then
I^(-1) = I

2. Prove that an inverse of an inverse of a matrix is this matrix itself.
In other words, if U is an invertible square matrix and matrix V is its inverse, then
V^(-1)=U

3. Prove that an inverse of a product of two invertible matrices equals to a product of their corresponding inverse matrices in a reverse order.
In other words, if X and Y are two invertible matrices then
(X·Y)^(-1) = Y^(-1)·X^(-1)
Expand this theorem to a case of any number of matrices.

4. What is the inverse of a square diagonal matrix with only elements along the main diagonal from top left corner to bottom right not equal to zero?

5. An inverse of a product of an invertible matrix by a non-zero scalar equals to a product of its inverse matrix by an inverted scalar.
In other words, if A is an invertible matrix and k is a non-zero real number, then
(k·A)^(-1) = (1/k)·A^(-1)

6. Prove that matrix operations of transposition and inversion commute for an invertible matrix.
In other words, if U is an invertible matrix, then
(U^T)^(-1) = (U^(-1))^T
Hint: Prove first that
(A·B)^T = A^T·B^T