Question
Find matrix \(A\) such that \[ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} A = \begin{bmatrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{bmatrix} \]
Solution
Step 1: Assume \(A\)
\[ A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \]Step 2: Multiply
\[ \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} = \begin{bmatrix} a+d & b+e & c+f \\ d & e & f \end{bmatrix} \]Step 3: Compare
\[ \begin{bmatrix} a+d & b+e & c+f \\ d & e & f \end{bmatrix} = \begin{bmatrix} 3 & 3 & 5 \\ 1 & 0 & 1 \end{bmatrix} \]Step 4: Solve
From second row: \[ d=1,\quad e=0,\quad f=1 \] From first row: \[ a+1=3 \Rightarrow a=2 \] \[ b+0=3 \Rightarrow b=3 \] \[ c+1=5 \Rightarrow c=4 \]Final Answer
\[
A =
\begin{bmatrix}
2 & 3 & 4 \\
1 & 0 & 1
\end{bmatrix}
\]