5.5. The Determinant of an Upper Triangular Matrix

The Gauss-Jordan elimination process is also one of the best methods for calculating determinants.  For convenience, we summarize the effects of the elementary row operations [§4.18] on the value of the determinant.

       

Operation

Effect

Proof

Interchange 2 rows

detA  ®  -detA

Theorem 5.3

Multiply a row by scalar t

detA  ®  t detA

Axiom 1

Add to one row a scalar

multiple of another

detA  ®  detA

Theorem 5.3

 

We shall show that the last operation alone is sufficient to transform an upper triangular matrix into a diagonal one.

Definition:  Upper Triangular Matrix

An upper triangular matrix is any n ´ n matrix of the form

        U=( u 11 u 12 u 1n 0 u 22 u 2n 0 0 u nn )                          (5.6)

so that all elements below the diagonal vanish, i.e., u ij =0     i>j .

Theorem 5.5.

For an n ´ n upper diagonal matrix U,

        detU= u 11 u 22 u nn   = i=1 n u ii                             (5.7)

Proof

It is easy to see that if we apply the backward substitution process of the Gauss-Jordan method [see §4.18] to (5.6), we can transform U to the equivalent form

U=diag( u 11 ,   u 22 ,,   u nn )

provided u ii 0     i .  Applying Theorem 5.4 then gives (5.7).  If u ii =0  for some i, the process breaks down because row i becomes a zero vector.  However, by Theorem 5.1, we have detU=0  so that (5.7) again holds.  QED.