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Let A be a 4×4 matrix and suppose that det(A)=3. For each of the following row operations, determine the value of det(B), where B is the matrix obtained by applying that row operation to A. a) Add 4 times row 4 to row 3 b) Multiply row 3 by 3 c) Interchange rows 1 and 4 Resulting values for det(B):

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steffaniasierrag

Answer:

a)3

b) 9

c) -3

Step-by-step explanation:

a) If B is obtained by adding a multiple of a row of A to another row of A, [tex]det (B) = det (A).[/tex]

Then, [tex]det(B)=3[/tex].

b) If B is obtained by multiplying a row of A by k, then [tex]det (B) = kdet (A)[/tex]. Then [tex]det(B)=3det(A)=3*3=9[/tex]

c) If B is obtained by exchanging two rows (columns) of A, then [tex]det (B) = - det (A)[/tex]. Then [tex]det (B) = - 3[/tex]

facundo3141592

By using determinant properties, we will see that:

  • A) det(B) = 3
  • B) det(B) = 9
  • C) det(B) = -3

How to find the determinant of the resulting matrices?

We know that A is a 4x4 matrix with:

det(A) = 3.

a) Here we are applying a transformation to A, such that we are adding one of the rows to another of the rows. These types of operations don't change the determinant of the matrix, so in this case:

det(B) = det(A) = 3

b) If B is obtained by multiplying a row of A by a factor K, then:

det(B) = K*det(A).

Then in this case, we have:

det(B) = 3*det(A) = 9

c) If we interchange two rows once (this is an odd permutation) then the sign of the determinant changes:

det(B) = -det(A) = -3

If you want to learn more about matrices, you can read:

https://brainly.com/question/11989522

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