19.2. AN INTRODUCTION TO DETERMINANTS 413

From the above corollary, it suffices to take the product of the diagonal elements. Thusdet(A) = 1×2×3× (−1) =−6. Without using the corollary, you could expand along thefirst column. This gives

1

∣∣∣∣∣∣2 6 70 3 33.70 0 −1

∣∣∣∣∣∣+0(−1)2+1

∣∣∣∣∣∣2 3 770 3 33.70 0 −1

∣∣∣∣∣∣+0(−1)3+1

∣∣∣∣∣∣2 3 772 6 70 0 −1

∣∣∣∣∣∣+0(−1)4+1

∣∣∣∣∣∣2 3 772 6 70 3 33.7

∣∣∣∣∣∣and the only nonzero term in the expansion is

1

∣∣∣∣∣∣2 6 70 3 33.70 0 −1

∣∣∣∣∣∣ .Now expand this along the first column to obtain

1×(

2×∣∣∣∣ 3 33.7

0 −1

∣∣∣∣+0(−1)2+1∣∣∣∣ 6 7

0 −1

∣∣∣∣+0(−1)3+1∣∣∣∣ 6 7

3 33.7

∣∣∣∣)= 1×2×

∣∣∣∣ 3 33.70 −1

∣∣∣∣Next expand this last determinant along the first column to obtain the above equals

1×2×3× (−1) =−6

which is just the product of the entries down the main diagonal of the original matrix. Itworks this way in general.

19.2.3 Properties of DeterminantsThere are many properties satisfied by determinants. Some of these properties have to dowith row operations. Recall the row operations.

Definition 19.2.15 The row operations consist of the following

1. Switch two rows.

2. Multiply a row by a nonzero number.

3. Replace a row by a multiple of another row added to itself.

Theorem 19.2.16 Let A be an n× n matrix and let A1 be a matrix which resultsfrom multiplying some row of A by a scalar c. Then cdet(A) = det(A1).

Example 19.2.17 Let A =

(1 23 4

),A1 =

(2 43 4

). det(A) =−2, det(A1) =−4.

Theorem 19.2.18 Let A be an n× n matrix and let A1 be a matrix which resultsfrom switching two rows of A. Then det(A) =−det(A1) . Also, if one row of A is a multipleof another row of A, then det(A) = 0.