Module 3 Quiz: Determinants
Quiz
10 questions on determinants, properties, Cramer's Rule, and geometric meaning.
1
det([4 -2; 6 1]) = ?
4(1) - (-2)(6) = 4 + 12 = 16.
2
What is the cofactor C_{12} of [2 3 1; 0 -1 4; 1 2 0]?
C_{12} = (-1)^{1+2} det([0 4;1 0]) = -(0-4) = 4.
3
How does a row swap affect the determinant?
It negates the determinant (multiplies by -1).
4
det(AB) = ? in terms of det(A) and det(B).
det(AB) = det(A) * det(B).
5
If A is 3x3 with det(A) = 5, what is det(A^{-1})?
det(A^{-1}) = 1/det(A) = 1/5.
6
Use Cramer's Rule: 2x + y = 4, x + 3y = 7.
det(A)=6-1=5. det(A_1)=det([4 1;7 3])=12-7=5. det(A_2)=det([2 4;1 7])=14-4=10. x=5/5=1, y=10/5=2.
7
What is the area of the parallelogram with sides (2,0) and (0,5)?
det([2 0;0 5]) = 10. Area = 10.
8
det of a triangular matrix = ?
Product of the diagonal entries.
9
A linear transformation has det = -2. What happens to area and orientation?
Areas are scaled by |det| = 2 (doubled). Orientation is reversed (negative det).
10
True or False: det(A + B) = det(A) + det(B).
False. The determinant is multiplicative (det(AB)=det(A)det(B)), not additive.