To find the matrix [tex]\( N \)[/tex] given that [tex]\( M = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \)[/tex] and [tex]\( M N = \begin{pmatrix} -6 & 3 \\ 4 & -2 \end{pmatrix} \)[/tex], we should solve for [tex]\( N \)[/tex] by considering the matrix operations involved.
1. Check if [tex]\( M \)[/tex] is invertible:
- First, we need to determine if matrix [tex]\( M \)[/tex] is invertible. For a matrix to be invertible, it must be square (number of rows equal to the number of columns) and have a non-zero determinant.
However, [tex]\( M \)[/tex] given here is a [tex]\( 2 \times 1 \)[/tex] matrix and not a square matrix. Thus, [tex]\( M \)[/tex] cannot be inverted by definition.
2. Solve for [tex]\( N \)[/tex]:
- Since [tex]\( M \)[/tex] is not invertible, [tex]\( N \)[/tex] cannot be solved by the typical method of multiplying [tex]\( \text{MN} \)[/tex] by [tex]\( M^{-1} \)[/tex], because [tex]\( M^{-1} \)[/tex] does not exist.
As a result:
- The matrix [tex]\( M \)[/tex] is not invertible.
- Because of the non-invertibility of [tex]\( M \)[/tex], we cannot find a unique matrix [tex]\( N \)[/tex] that satisfies the given equation [tex]\( M N \)[/tex].
Therefore, the solution to find the matrix [tex]\( N \)[/tex] is not possible with the information given. Thus, there is no solution for matrix [tex]\( N \)[/tex] under these conditions.
