Question

The square matrix A is called orthogonal provided

thatAT=A-1. Show that the determinant of such
amatrix must be either +1 or -1.

Answer 1

Answer: The proof is done below.

Step-by-step explanation: Given that the square matrix A is called orthogonal provided that
`A^T=A^(-1).`

We are to show that the determinant of such a matrix is either +1 or -1.

We will be using the following result :

`|A^(-1)|=(1)/(|A|).`

Given that, for matrix A,

`A^T=A^(-1).`

Taking determinant of the matrices on both sides of the above equation, we get

`|A^T|=|A^(-1)|\\\\\Rightarrow |A|=(1)/(|A|)~~~~~~~~~~~~~~~~~~~~[\textup{since A and its transpose have same determinant}]\\\\\\\Rightarrow |A|^2=1\\\\\Rightarrow |A|=\pm1~~~~~~~~~~~~~~~~~~~~[\textup{taking square root on both sides}]`

Thus, the determinant of matrix A is either +1 or -1.

Hence showed.