Let A = a b c d and let k be a scalar. Find a formula that relates det(kA) to k and det(A). Find det(A). det(A) = (Simplify your answer.)
Let A = a b c d and let k be a scalar. Find a formula that relates det(kA) to k and det(A). Find det(A). det(A) = (Simplify your answer.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Hello there! Can you help me solve a problem with subparts? Thanks!
![**Problem Statement**
Let \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) and let \( k \) be a scalar. Find a formula that relates \( \text{det}(kA) \) to \( k \) and \( \text{det}(A) \).
---
**Steps to Solve**
1. **Find \(\text{det}(A)\):**
The determinant of matrix \( A \) is calculated as:
\[
\text{det}(A) = ad - bc
\]
(Simplify your answer.)
**Explanation**
To solve this problem, you need to understand how the determinant of a scalar multiple of a matrix, \( kA \), relates to \( k \) and the determinant of the original matrix, \( A \).
For a \( 2 \times 2 \) matrix, if you multiply the entire matrix by a scalar \( k \), the determinant of the resulting matrix \( kA \) is:
\[
\text{det}(kA) = k^2 \cdot \text{det}(A)
\]
This is because each row (or column) of the matrix is multiplied by the scalar \( k \), and since a determinant of a matrix involves the product of the elements of its rows and columns, the scalar appears as a multiplicative factor for each row (or column). For a \( 2 \times 2 \) matrix, this leads to the scalar being squared.
Use this relationship to determine how changing the scalar \( k \) affects the determinant of the matrix \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2F83da9d7f-4e60-44c4-8733-6c235daee470%2Fxelznxp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Let \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) and let \( k \) be a scalar. Find a formula that relates \( \text{det}(kA) \) to \( k \) and \( \text{det}(A) \).
---
**Steps to Solve**
1. **Find \(\text{det}(A)\):**
The determinant of matrix \( A \) is calculated as:
\[
\text{det}(A) = ad - bc
\]
(Simplify your answer.)
**Explanation**
To solve this problem, you need to understand how the determinant of a scalar multiple of a matrix, \( kA \), relates to \( k \) and the determinant of the original matrix, \( A \).
For a \( 2 \times 2 \) matrix, if you multiply the entire matrix by a scalar \( k \), the determinant of the resulting matrix \( kA \) is:
\[
\text{det}(kA) = k^2 \cdot \text{det}(A)
\]
This is because each row (or column) of the matrix is multiplied by the scalar \( k \), and since a determinant of a matrix involves the product of the elements of its rows and columns, the scalar appears as a multiplicative factor for each row (or column). For a \( 2 \times 2 \) matrix, this leads to the scalar being squared.
Use this relationship to determine how changing the scalar \( k \) affects the determinant of the matrix \( A \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
