The value of the determinant is
A)60.
B)6.
C)54.
D)21.
E)None of the above

Question

The value of the determinant  is
A)60. 
B)6. 
C)54. 
D)21. 
E)None of the above

Quizplus · Accepted Answer

We can use the following properties of determinants:
- Swapping two rows multiplies the determinant by -1.
- Multiplying a row by a scalar multiplies the determinant by that scalar.
- Adding a multiple of one row to another row does not change the determinant.

Using these properties, we can simplify the given determinant as follows:

11eabb83_5a15_5a98_b8db_916ef4958d9f_TB2950_11
= 11eabb83_5a15_5a98_b8db_916ef4958d9f_TB2950_11/11 + 10eabb83_5a15_5a98_b8db_916ef4958d9f_TB2950_11/11
= eabb83_5a15_5a98_b8db_916ef4958d9f_TB2950_11 + 10eabb83_5a15_5a98_b8db_916ef4958d9f_TB2950_11/11

Now, we can use the fact that the determinant of a 2x2 matrix [a b; c d] is ad - bc, to evaluate the determinant of the matrix formed by the first two rows of the given determinant:

eabb83_5a15_5a98_b8db_916ef4958d9f
10eabb83_5a15_5a98_b8db_916ef4958d9f

This determinant is equal to eabb83_5a15_5a98_b8db_916ef4958d9f * 10 - 10eabb83_5a15_5a98_b8db_916ef4958d9f = -90eabb83_5a15_5a98_b8db_916ef4958d9f.

Finally, we can use the fact that multiplying any row of a matrix by -1 multiplies the determinant by -1, to evaluate the determinant of the original matrix:

-(-90eabb83_5a15_5a98_b8db_916ef4958d9f) + TB2950_11 * (-10eabb83_5a15_5a98_b8db_916ef4958d9f/11)
= 90eabb83_5a15_5a98_b8db_916ef4958d9f - TB2950_11 * eabb83_5a15_5a98_b8db_916ef4958d9f/11

We can see that the determinant is divisible by 6 and 9, so it is also divisible by 54. Checking the answer choices, we see that the only option that is divisible by 54 is C, so the answer is C.

The Value of the Determinant Is