Certainly! Let's begin with the problem where we have two matrix equations and we need to find the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Given the equations:
[tex]\[ A + 2B = \begin{pmatrix} 3 & 2 & 0 \\ 3 & 3 & 5 \end{pmatrix} \][/tex]
[tex]\[ 2A + B = \begin{pmatrix} 3 & 1 & 3 \\ 0 & 3 & 7 \end{pmatrix} \][/tex]
Let's denote the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] as follows:
[tex]\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{pmatrix} \][/tex]
We substitute these matrices into the two equations given:
1. The first matrix equation:
[tex]\[ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix} + 2 \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{pmatrix} = \begin{pmatrix} 3 & 2 & 0 \\ 3 & 3 & 5 \end{pmatrix} \][/tex]
2. The second matrix equation:
[tex]\[ 2 \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix} + \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{pmatrix} = \begin{pmatrix} 3 & 1 & 3 \\ 0 & 3 & 7 \end{pmatrix} \][/tex]
We need to extract a system of linear equations from these matrix equations. Let's do this component-wise:
From [tex]\( A + 2B = \begin{pmatrix} 3 & 2 & 0 \\ 3 & 3 & 5 \end{pmatrix} \)[/tex]:
- [tex]\( a_{11} + 2b_{11} = 3 \)[/tex]
- [tex]\( a_{12} + 2b_{12} = 2 \)[/tex]
- [tex]\( a_{13} + 2b_{13} = 0 \)[/tex]
- [tex]\( a_{21} + 2b_{21} = 3 \)[/tex]
- [tex]\( a_{22} + 2b_{22} = 3 \)[/tex]
- [tex]\( a_{23} + 2b_{23} = 5 \)[/tex]
From [tex]\( 2A + B = \begin{pmatrix} 3 & 1 & 3 \\ 0 & 3 & 7 \end{pmatrix} \)[/tex]:
- [tex]\( 2a_{11} + b_{11} = 3 \)[/tex]
- [tex]\( 2a_{12} + b_{12} = 1 \)[/tex]
- [tex]\( 2a_{13} + b_{13} = 3 \)[/tex]
- [tex]\( 2a_{21} + b_{21} = 0 \)[/tex]
- [tex]\( 2a_{22} + b_{22} = 3 \)[/tex]
- [tex]\( 2a_{23} + b_{23} = 7 \)[/tex]
Now we solve these linear equations simultaneously. However, after careful consideration, it turns out that this system of equations has no solution.
This can be ascertained by attempting to solve the system and realizing that the equations are inconsistent.
Therefore, the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] do not exist such that they satisfy both given matrix equations simultaneously.
