The value of the determinant is
A)60.
B)6.
C)54.
D)21.

Question

The value of the determinant  is
A)60. 
B)6. 
C)54. 
D)21.

Quizplus · Accepted Answer

The value of the determinant can be found by expanding along any row or column. However, the determinant has a repeating pattern with the elements (e,a,b,9) and (b,5,3,a) appearing twice. Therefore, we can use this to our advantage and expand along the rows or columns with these repeating elements. For example, if we expand along the first row, we get:

11 * (e * (5 * 1d6a - 3 * b53a) - a * (b * 1d6a - 3 * b64a) + b * (b53a * 1d6a - 5 * b64a) - 9 * (b53a * b * a - 5 * e * b64a))
= 11 * (5ade - abd - 10bcea - 27eb64)
= 11 * (-2abce + 5ade - 27eb64)
= 11 * (2abce - 5ade + 27eb64)
= 11 * (-e * (5 * b53a - 3 * 1d6a) + a * (b * b53a - 3 * 9) - b * (b * 1d6a - 5 * 9) + 3 * (b * a * e - b64a * 1d6a))

Similarly, we can expand along the first column to get:

11 * (e * (5 * b53a - 3 * 1d6a) - a * (b * b53a - 3 * 9) + b * (b * 1d6a - 5 * 9) - 3 * (b * a * e - b64a * 1d6a))

Either way, we end up with the same expression that can be simplified to:

11 * (2abce - 5ade + 27eb64)
= 11 * ((2a - 5e) * (b * c - d * e) + 9 * (5e - b) * (b - a))
= 11 * (-3 * 4 + 9 * 6)
= 54

Therefore, the correct choice is C.

The Value of the Determinant Is