Multiplying a determinant by a constant? book says

In summary, multiplying a determinant by a constant allows us to scale the entire matrix, resulting in a new determinant with different values. This new determinant's value will be the original determinant multiplied by the constant. We can multiply a determinant by any constant, and it will not change the determinant's properties. There is a specific rule for multiplying a determinant by a constant, which states that the constant can be distributed to each element inside the determinant.
  • #1
mr_coffee
1,629
1
I'm confused, this question had 5 parts and i got the other 4 but this one I keep missing...
If A is a 2x2 matrix...
Det(A) = -5;
The Det(-3A);
The book said, if u multiply a column by a constant k then the determinant is also multiplied by k. So wouldn't the answer just be (-5)(-3) = 15?
Det(-3A) = 15? it says its wrong htough...:bugeye:
 
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  • #2
Evaluate [tex]\left|\begin{array}{cc} ka& kb \\ kc&kd\end{array}\right| [/tex]... and compare it to [tex]\left|\begin{array}{cc} a& b \\ c&d\end{array}\right| [/tex]. Next try the 3x3 case.
 
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Related to Multiplying a determinant by a constant? book says

1. What is the purpose of multiplying a determinant by a constant?

Multiplying a determinant by a constant allows us to scale the entire matrix, resulting in a new determinant with different values.

2. How does multiplying a determinant by a constant affect its value?

Multiplying a determinant by a constant will result in a new determinant with a value that is the original determinant multiplied by the constant. For example, if the original determinant's value is 5 and we multiply it by 2, the new determinant's value will be 10.

3. Can we multiply a determinant by any constant?

Yes, we can multiply a determinant by any constant, whether it is a positive number, negative number, or zero. This will result in a new determinant with the same number of rows and columns, but with different values.

4. How does multiplying a determinant by a constant affect the determinant's properties?

Multiplying a determinant by a constant does not change its properties. The determinant will still have the same number of rows and columns, and it will still be an array of numbers. However, the values of the determinant will change according to the constant used.

5. Is there a specific rule for multiplying a determinant by a constant?

Yes, there is a specific rule for multiplying a determinant by a constant. The rule states that when a constant is multiplied by a determinant, it can be placed outside of the determinant and distributed to each element inside the determinant. For example, if we have a determinant [a b; c d] and we multiply it by a constant k, the new determinant will be [ka kb; kc kd].

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