The discussion centers on calculating the determinant of a 3x3 matrix using row operations. The matrix in question is: 1 -2 2; 0 5 -1; 2 -4 1. The user initially calculated the determinant as -30 after performing row operations, but the correct determinant is -15. The confusion arises from the distinction between adding a multiple of a row to another row, which does not change the determinant, and multiplying a row by a scalar, which does. The relevant theorem states that interchanging rows changes the sign of the determinant, multiplying a row by k multiplies the determinant by k, and adding a multiple of one row to another does not affect the determinant.
Students studying linear algebra, educators teaching matrix theory, and anyone looking to deepen their understanding of determinants and row operations.
You did not replace a row by some multiple of itself; you added a multiple of a row to another row. These are different operations. On the other hand, if you had replaced row 1 by -2 times itself, and then added the first row to the third row, then your determinant would have been +30. This is not what you did though, since the first row stayed the same from start to finish.vs55 said:hmmm well i looked over my book, and it said if i multiply a row by k...i multiply the determinant by k...so when do we multiply the determinant by k?..cause what i did was that not multiplying a row by k(2)?
and thanks for ur help guys