Find a and b such that 2 1 20 = a 2 +b -4 -22 -1 a = b =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Matrix Equation Problem**
The task is to find the values of \( a \) and \( b \) such that:
\[
\begin{bmatrix}
2 \\
20 \\
-22
\end{bmatrix}
=
a \begin{bmatrix}
1 \\
2 \\
-1
\end{bmatrix}
+
b \begin{bmatrix}
0 \\
-4 \\
5
\end{bmatrix}
\]
**Steps to Solve:**
1. Set up the system of linear equations based on the matrices:
- \( a \times 1 + b \times 0 = 2 \)
- \( a \times 2 + b \times (-4) = 20 \)
- \( a \times (-1) + b \times 5 = -22 \)
2. Solve the first equation for \( a \):
- \( a = 2 \)
3. Substitute \( a = 2 \) into the second equation:
- \( 2 \times 2 - 4b = 20 \)
- Simplify to find \( b \).
4. Substitute \( a = 2 \) into the third equation if necessary to verify \( b \).
**Solution:**
- \( a = \) [answer derived from solving]
- \( b = \) [answer derived from solving]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70a05657-0307-4b7c-962c-ab3824846297%2F3154a90c-0f2f-442c-bb84-686d9d5add92%2Fjp8pqwp_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Equation Problem**
The task is to find the values of \( a \) and \( b \) such that:
\[
\begin{bmatrix}
2 \\
20 \\
-22
\end{bmatrix}
=
a \begin{bmatrix}
1 \\
2 \\
-1
\end{bmatrix}
+
b \begin{bmatrix}
0 \\
-4 \\
5
\end{bmatrix}
\]
**Steps to Solve:**
1. Set up the system of linear equations based on the matrices:
- \( a \times 1 + b \times 0 = 2 \)
- \( a \times 2 + b \times (-4) = 20 \)
- \( a \times (-1) + b \times 5 = -22 \)
2. Solve the first equation for \( a \):
- \( a = 2 \)
3. Substitute \( a = 2 \) into the second equation:
- \( 2 \times 2 - 4b = 20 \)
- Simplify to find \( b \).
4. Substitute \( a = 2 \) into the third equation if necessary to verify \( b \).
**Solution:**
- \( a = \) [answer derived from solving]
- \( b = \) [answer derived from solving]
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