Scaling a row scales the determinant

In summary, the conversation discusses a homework problem involving scaling and adding rows to a determinant. The attempted solution suggests using a scaling factor of -2 and adding multiples of row 1 to rows 2 and 3. However, the gaps between the numbers in column 1 and 3 do not match after performing the row operations. A hint is given to consider the gaps between the numbers in the original and modified matrices.
  • #1
Lifprasir
16
0

Homework Statement


http://puu.sh/1rcsO

I got the first one from a simple scaling, but I can not figure out the second one.

Homework Equations


Det(cA) = cDet(A)
Scaling a row scales the determinant
Adding rows/columns to each other does not affect the determinant
Det(AT) = Det A

The Attempt at a Solution


So, shouldn't I be able to scale http://puu.sh/1rcyZ by -2, and just add row 2 and row 3 by multiples of row 1 ? I actually don't even think this is even close to right because the column 1 and 3 won't resemble each other at all once the row operations are done...
 
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  • #2
Hi Lifprasir! :smile:

Hint: what are the gaps between 1,2,3?

what are the gaps between -2,-8,-14? :wink:
 

1. How does scaling a row affect the determinant?

Scaling a row of a matrix by a constant k multiplies the determinant by the same constant k. This means that the determinant will either increase or decrease depending on the value of k, but its sign will remain the same.

2. Can scaling a row change the sign of the determinant?

No, scaling a row by a positive or negative constant will not change the sign of the determinant. The sign of the determinant is determined by the number of row interchanges required to reduce the matrix to upper triangular form.

3. Is scaling a row the only operation that affects the determinant?

No, there are other elementary row operations that can affect the determinant, such as swapping two rows or adding a multiple of one row to another. However, scaling a row is a simple and efficient way to manipulate the determinant.

4. How does scaling a row affect the solutions of a system of linear equations?

Scaling a row does not change the solutions of a system of linear equations. This is because the same constant is applied to both the coefficients and the constants in the equations, resulting in equivalent equations and the same solutions.

5. What is the purpose of scaling a row when calculating the determinant?

Scaling a row allows us to simplify the matrix and make it easier to calculate the determinant. By scaling a row, we can create zeros in specific positions, which can then be eliminated through row operations to reduce the matrix to upper triangular form. This makes it easier to calculate the determinant using a method like cofactor expansion.

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