Show that det In = 1

Question

Show that det In = 1

Answer 1

I_n is a diagonal matrix with all ones along the diagonal and zeroes elsewhere. If you expand det(I_n) using the first row, you have:

 

det(I_n) = (-1)¹⁺¹(1) det(I_n-1) = det(I_n-1). This is true since the minor of I_n from the (1,1) element is I_n-1.

 

Now, employ mathematical induction to construct the rest of the proof:

 

1) det(I₁) = 1  (I₁ is just the real number 1.)

2) det(I_n) = det(I_n-1) for n > 1. (This was just demonstrated above.)

 

By mathematical induction, det(I_n) = 1 for all positive integers n.

 

(If you don't know about mathematical induction yet, your math education is severely impoverished.)