2. Use Elementary Operations (a) Compute the determinant of the matrix A, reduci

Question

2. Use Elementary Operations (a) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: 2245 2223 (b) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: (c) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: 0 0 01 0 0 1 1 (d) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations (e) Compute the determinant of the matrix A, reducing the matrix to a simpler matrix (usually triangular), by elementary operations: 0 0 0 1

Explanation / Answer

2. We know that

(a)We will reduce A to to a triangular matrix as under:

1.Add 3 times the 1st row to the 2nd row

2.Add -2 times the 1st row to the 3rd row

3.Add -2 times the 1st row to the 4th row

4.Multiply the 3rd row by 1/2

5.Add -3/2 times the 4th row to the 3rd row

Then A changes to B =

1

1

1

1

0

1

1

1

0

0

1

0

0

0

0

1

Now, B is an upper triangular matrix , so that det(B) = 1*1*1*1 = 1. Further, only the 4th row operation has the effect of scaling the value of det(A) by ½ so that det(BG) = ½ det(A). Hence det(A) = 2det(B) = 2.

       (b). If we interchange the 1st and the 4th rows, then A changes to B =

1

1

1

1

0

1

1

1

0

0

1

1

0

0

0

1

It may be observed that B is an upper triangular matrix so that det(B) = 1*1*1*1 = 1. Further, in arriving at the matrix B, there is only one row operation, namely change of the 1st and the 4th rows. Hence det(B) = -det(A) or, det(A) = -det(B) = -1.

(c ).We will reduce A to to a triangular matrix as under:

Then A changes to B =

1

1

1

1

0

1

1

1

0

0

1

1

0

0

0

1

It may be observed that B is an upper triangular matrix so that det(B) = 1*1*1*1 = 1. Further, the 1st row operation has the effect of changing the sign of det(A) while the 2nd row operation has the effect of scaling the value of det(A) by 1/2. Hence det(B) = (-1/2)det(A) so that det(A) = (-2)det(B) = -2.

       (d). We will reduce A to to a triangular matrix as under:

            Then A changes to B =

1

1

1

1

0

1

1

1

0

0

1

1

0

0

0

1

Again, det(B) = 1*1*1*1 = 1, Also, det(B) = (-1/2)det(A) so that det(A) = -2 det(B)= -2.

         (e ). We will reduce A to to a triangular matrix as under:

Then A changes to B =

1

1

1

1

0

1

1

1

0

0

1

1

0

0

0

1

            Again, det(B) = 1*1*1*1 = 1, Also, det(B) = [- 1/(2)] det(A) so that det(A) = -(2)det(B) = -2

1

1

1

1

0

1

1

1

0

0

1

0

0

0

0

1

Related Questions

Navigate

• Browse (All)
• Browse 2
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.