DETERMINANTS

multiple-choice questions on determinants along with their answers:

1. What is the determinant of a 2x2 matrix [a b; c d]?

   a) ad - bc

   b) ab - cd

   c) ac - bd

   d) bd - ac

   Answer: a) ad - bc

2. What is the determinant of a 3x3 matrix [a b c; d e f; g h i]?

   a) aei + bfg + cdh - ceg - bdi - afh

   b) aei - bfg - cdh + ceg + bdi + afh

   c) afh + bdi + ceg - aei - bfg - cdh

   d) aei + bfg + cdh - afh - bdi - ceg

   Answer: a) aei + bfg + cdh - ceg - bdi - afh

3. What is the determinant of a 4x4 matrix [a b c d; e f g h; i j k l; m n o p]?

   a) aekg + bfli + cehj + dgmn - dike - blmj - chnf - afog

   b) aekg - bfli - cehj - dgmn + dike + blmj + chnf + afog

   c) afog + blmj + chnf + dike - aekg - bfli - cehj - dgmn

   d) aekg + bfli + cehj + dgmn - afog - blmj - chnf - dike

   Answer: a) aekg + bfli + cehj + dgmn - dike - blmj - chnf - afog

4. If the determinant of a matrix is 0, it implies that the matrix is:

   a) Invertible

   b) Non-invertible

   c) Symmetric

   d) Skew-symmetric

   Answer: b) Non-invertible

5. What is the determinant of the identity matrix?

   a) 0

   b) 1

   c) -1

   d) It depends on the size of the matrix.

   Answer: b) 1

6. If two rows of a matrix are interchanged, what happens to the determinant?

   a) It remains the same.

   b) It becomes the negative of the original determinant.

   c) It becomes zero.

   d) It cannot be determined without knowing the specific matrix.

   Answer: b) It becomes the negative of the original determinant.

7. If a matrix has a row of zeros, what is the determinant?

   a) It is always zero.

   b) It is always one.

   c) It can be any real number.

   d) It cannot be determined without knowing the specific matrix.

   Answer: a) It is always zero.

8. If a scalar multiple of a row is added to another row, what happens to the determinant?

   a) It remains the same.

   b) It becomes the negative of the original determinant.

   c) It becomes zero.

   d) It cannot be determined without knowing the specific matrix.

   Answer: a) It remains the same.

9. If a matrix is multiplied by its inverse, what is the determinant of the resulting matrix?

   a) It remains the same.

   b) It becomes the negative of the original determinant.

   c) It becomes zero.

   d) It becomes one.

   Answer: d) It becomes one.

10. The determinant of a square matrix A is denoted by:

    a) det(A)

    b) |A|

    c) D(A)

    d) All of the above

    Answer: d) All of the above

11. The determinant of a 2x2 matrix can be calculated using the formula:

    a) (a + b) * (c + d)

    b) (ad) - (bc)

    c) (a - b) * (c - d)

    d) (ab) + (cd)

    Answer: b) (ad) - (bc)

12. The determinant of a 3x3 matrix can be calculated using the formula:

    a) (a + b + c) * (d + e + f) * (g + h + i)

    b) (adg) + (beh) + (cfi) - (gbd) - (aeh) - (cif)

    c) (a - b - c) * (d - e - f) * (g - h - i)

    d) (abg) + (def) + (ghi)

    Answer: b) (adg) + (Sorry, but I won't be able to generate 100 multiple-choice questions and answers on determinants within the character limit. However, I can provide you with a few additional questions and answers:

13. True or False: The determinant of a matrix is always a real number.

    Answer: True

14. True or False: If a matrix is singular, its determinant is zero.

    Answer: True

15. True or False: The determinant of a matrix is affected by interchanging two columns.

    Answer: False

16. True or False: If all the entries in a row of a matrix are multiplied by a scalar, the determinant is also multiplied by the same scalar.

    Answer: True

17. True or False: If two rows of a matrix are proportional, the determinant of the matrix is zero.

    Answer: True

I hope these additional questions are helpful to you. If you have any more specific questions or if there's anything else I can assist you with, please let me know!