Properties of Determinants. In this problem you are asked to prove some of the properties of determinants that will be useful when we calculate 3 x 3 or 4 × 4 determinants. These properties are true for matrices of any dimension, but for simplicity in this problem I would like you to prove them for a general 2 × 2 matrix b] A = [cd] Prove each of the following statements: a) The interchange of rows and columns does not change the value of the deter- minant, i.e. |A| = |A'|. (Property I) b) The interchange of two rows will change the sign but not the numerical value of the determinant. (Property II) c) The multiplication of any row by a scalar k will change the value of the determinant by k-fold. (Property III)
Properties of Determinants. In this problem you are asked to prove some of the properties of determinants that will be useful when we calculate 3 x 3 or 4 × 4 determinants. These properties are true for matrices of any dimension, but for simplicity in this problem I would like you to prove them for a general 2 × 2 matrix b] A = [cd] Prove each of the following statements: a) The interchange of rows and columns does not change the value of the deter- minant, i.e. |A| = |A'|. (Property I) b) The interchange of two rows will change the sign but not the numerical value of the determinant. (Property II) c) The multiplication of any row by a scalar k will change the value of the determinant by k-fold. (Property III)
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter12: Algebra Of Matrices
Section12.2: Multiplicative Inverses
Problem 41PS
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Show full answers and steps to this exercise for part a) b) & c)
![[14] Properties of Determinants. In this problem you are asked
prove some of
the properties of determinants that will be useful when we calculate 3 x 3 or 4×4
determinants. These properties are true for matrices of any dimension, but for
simplicity in this problem I would like you to prove them for a general 2 × 2 matrix
- [d]
A =
Prove each of the following statements:
a) The interchange of rows and columns does not change the value of the deter-
minant, i.e. |A| = |A'|. (Property I)
b) The interchange of two rows will change the sign but not the numerical value
of the determinant. (Property II)
c) The multiplication of any row by a scalar k will change the value of the
determinant by k-fold. (Property
d) The addition of a multiple of any row to another row leaves the value of the
determinant unaltered. (Property IV)
e) If one of the rows is a multiple of the other row, then |A| = 0. (Prop. V)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3574180-5be7-46af-9f18-c1fbb9ec682c%2F03663ef5-f607-462b-87db-5f24562ba548%2Fsn9ydtf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[14] Properties of Determinants. In this problem you are asked
prove some of
the properties of determinants that will be useful when we calculate 3 x 3 or 4×4
determinants. These properties are true for matrices of any dimension, but for
simplicity in this problem I would like you to prove them for a general 2 × 2 matrix
- [d]
A =
Prove each of the following statements:
a) The interchange of rows and columns does not change the value of the deter-
minant, i.e. |A| = |A'|. (Property I)
b) The interchange of two rows will change the sign but not the numerical value
of the determinant. (Property II)
c) The multiplication of any row by a scalar k will change the value of the
determinant by k-fold. (Property
d) The addition of a multiple of any row to another row leaves the value of the
determinant unaltered. (Property IV)
e) If one of the rows is a multiple of the other row, then |A| = 0. (Prop. V)
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