5.4. The Determinant of a Diagonal Matrix

Definition:  Diagonal Matrix

An n ´ n matrix A=( a ij )  is called a diagonal matrix if a ij =0  whenever ij .  In other words, it has the form

        A=( a 11 0 0 0 a 22 0 0 0 a nn )

and can be denoted

A=diag( a 11 , a 22 ,, a nn )

Theorem 5.4.

If A=diag( a 11 , a 22 ,, a nn ) , then

        detA= a 11 a 22 a nn   = i=1 n a ii                                     (5.4)

Proof

Using Ij to denote the ith unit coordinate vector, we can write A as

        A=( a 11 I 1 ,   a 22 I 2 ,  ,   a nn I n )

Applying axiom 1 repeatedly, we have

        detA=d( a 11 I 1 ,   a 22 I 2 ,  ,   a nn I n )   = a 11 d( I 1 ,   a 22 I 2 ,  ,   a nn I n )

                = a 11 a 22 d( I 1 ,   I 2 ,  ,   a nn I n )   = a 11 a 22 a nn d( I 1 ,   I 2 ,  ,   I n )

                = a 11 a 22 a nn detI                                            (5.5)

                = a 11 a 22 a nn                    [ axiom 3 used ]