Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant. a b a + kc b+ kd c d d What is the elementary row operation? O A. Row 1 is replaced with the sum of itself and k times row 2. B. Row 2 is replaced with the sum of itself and k times row 1. C. Rows 1 and 2 are interchanged. D. Row 1 is multiplied by k. O E. Row 2 is multiplied by k. How does the row operation affect the determinant? O A. It changes the sign of the determinant. O B. It increases the determinant by k. OC. It multiplies the determinant by k.
Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant. a b a + kc b+ kd c d d What is the elementary row operation? O A. Row 1 is replaced with the sum of itself and k times row 2. B. Row 2 is replaced with the sum of itself and k times row 1. C. Rows 1 and 2 are interchanged. D. Row 1 is multiplied by k. O E. Row 2 is multiplied by k. How does the row operation affect the determinant? O A. It changes the sign of the determinant. O B. It increases the determinant by k. OC. It multiplies the determinant by k.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![## Exploring the Effect of an Elementary Row Operation on the Determinant of a Matrix
Consider the following matrix transformation:
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\rightarrow
\begin{pmatrix}
a + kc & b + kd \\
c & d
\end{pmatrix}
\]
### What is the Elementary Row Operation?
The transformation involves an elementary row operation. Which of the following operations is applied to the matrix?
- **A.** Row 1 is replaced with the sum of itself and \(k\) times row 2.
- **B.** Row 2 is replaced with the sum of itself and \(k\) times row 1.
- **C.** Rows 1 and 2 are interchanged.
- **D.** Row 1 is multiplied by \(k\).
- **E.** Row 2 is multiplied by \(k\).
### How Does the Row Operation Affect the Determinant?
Determine how the row operation affects the determinant of the matrix.
- **A.** It changes the sign of the determinant.
- **B.** It increases the determinant by \(k\).
- **C.** It multiplies the determinant by \(k\).
- **D.** It does not change the determinant.
By exploring the specific row operation and its impact, we can better understand the properties of matrix determinants and the effects of different transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2F1d343f37-8e7d-4393-9193-0d8d034ef793%2F8g8vwr_processed.png&w=3840&q=75)
Transcribed Image Text:## Exploring the Effect of an Elementary Row Operation on the Determinant of a Matrix
Consider the following matrix transformation:
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\rightarrow
\begin{pmatrix}
a + kc & b + kd \\
c & d
\end{pmatrix}
\]
### What is the Elementary Row Operation?
The transformation involves an elementary row operation. Which of the following operations is applied to the matrix?
- **A.** Row 1 is replaced with the sum of itself and \(k\) times row 2.
- **B.** Row 2 is replaced with the sum of itself and \(k\) times row 1.
- **C.** Rows 1 and 2 are interchanged.
- **D.** Row 1 is multiplied by \(k\).
- **E.** Row 2 is multiplied by \(k\).
### How Does the Row Operation Affect the Determinant?
Determine how the row operation affects the determinant of the matrix.
- **A.** It changes the sign of the determinant.
- **B.** It increases the determinant by \(k\).
- **C.** It multiplies the determinant by \(k\).
- **D.** It does not change the determinant.
By exploring the specific row operation and its impact, we can better understand the properties of matrix determinants and the effects of different transformations.
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