University of Aarhus
Department of Mathematical Sciences
Calculus Quiz
week 38.1
Legend: A 4 indicates a correct response; a 8, indicates an incorrect response, in this case, the correct answer is marked with a l. Correct and follow
a 4,l for a solution. Return through n.
Fall 2004
2
Determinants
1. The determinant
Yes
a b c d = ad − bc
No
2. The determinant
Yes
1
4
1
2
5
2
3
6 6= 0
3
No
3. If the product of square matrices AB = I, then the determinant:
|A| = −|B|
|A| =
1
|B|
3
4. If the determinant of a square matrix is nonzero, the the matrix is invertible.
Yes
No
5. If the coefficient matrix of the homogeneous system of equations Ax =
0 is a square matrix with determinant |A| = 0, then a solution is uniquely
given.
Yes
No
6. A row operation on a square matrix allways:
Leaves the determinant unchanged
Changes sign on the determinant
Multiplies the determinant with some nonzero number
Solutions to Quizzes
4
Solutions to Quizzes
Solution to Quiz:
Yes, by [LA] Section 8.
Solutions to Quizzes
5
Solution to Quiz:
No,
1 2 3
4 5 6 = 0
1 2 3
since row one and three are equal. See [LA] Section 8.
Solutions to Quizzes
6
Solution to Quiz:
By [LA] Theorem 11 “the product rule”:
|A||B| = |I| = 1
I.e.
|A| =
1
|B|
Solutions to Quizzes
Solution to Quiz:
Yes, this is [LA] Theorem 12.
7
Solutions to Quizzes
Solution to Quiz:
No, this is [LA] Theorem 13.
8
Solutions to Quizzes
9
Solution to Quiz:
[LA] Section 8:
• Interchanging of two rows: changes sign
• Scaling of a row : scales with same number
• Adding a scaled row: leaves fixed