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Find a and b such that 2 1 20 = a 2 +b -4 -22 -1 a = b =

Find a and b such that 2 1 20 = a 2 +b -4 -22 -1 a = b =

Advanced Engineering Mathematics
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]
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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