19.2. AN INTRODUCTION TO DETERMINANTS 415

Replace the second row by (−5) times the first row added to it. Then replace the thirdrow by (−4) times the first row added to it. Finally, replace the fourth row by (−2) timesthe first row added to it. This yields the matrix

B =

1 2 3 40 −9 −13 −170 −3 −8 −130 −2 −10 −3

and from Theorem 19.2.20, it has the same determinant as A. Now using other row opera-tions, det(B) =

(−13

)det(C) where

C =

1 2 3 40 0 11 220 −3 −8 −130 6 30 9

 .

The second row was replaced by (−3) times the third row added to the second row. ByTheorem 19.2.20 this didn’t change the value of the determinant. Then the last row wasmultiplied by (−3) . By Theorem 19.2.16 the resulting matrix has a determinant which is(−3) times the determinant of the un-multiplied matrix. Therefore, we multiplied by −1/3to retain the correct value. Now replace the last row with 2 times the third added to it.This does not change the value of the determinant by Theorem 19.2.20. Finally switchthe third and second rows. This causes the determinant to be multiplied by (−1) . Thusdet(C) =−det(D) where

D =

1 2 3 40 −3 −8 −130 0 11 220 0 14 −17

You could do more row operations or you could note that this can be easily expanded alongthe first column followed by expanding the 3×3 matrix which results along its first column.Thus

det(D) = 1(−3)∣∣∣∣ 11 22

14 −17

∣∣∣∣= 1485

and so det(C) =−1485 and det(A) = det(B) =(−1

3

)(−1485) = 495.

Example 19.2.26 Find the determinant of the matrix1 2 3 21 −3 2 12 1 2 53 −4 1 2

Replace the second row by (−1) times the first row added to it. Next take −2 times the

first row and add to the third and finally take −3 times the first row and add to the last row.This yields 

1 2 3 20 −5 −1 −10 −3 −4 10 −10 −8 −4

 .