Week 1 Q12

C12 Solve the given vector equation for a, or explain why no solution exists:

\[\begin{equation} \text{a} \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} + 4 \begin{bmatrix} 3 \\ 4 \\ 2 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \\ 4 \end{bmatrix} \end{equation}\]

Step 1:

Apply a and 4 to the vectors:

\[\begin{equation} (a*1) + (4*3) = -1 \\ (a*2) + (4*4) = 0 \\ (a*-1) + (4*2) = 4 \\ \end{equation}\]

Step 2:

Solve the equations for a:

\[\begin{equation} a = -13 \\ a = -8 \\ a = 4 \\ \end{equation}\]

Conclusion: This vector equation gives three individual answers for a, indicating that there is no solution to this vector equation.