Answer :
Certainly! Let's solve for matrix [tex]\( N \)[/tex] step by step given the matrices [tex]\( M \)[/tex] and [tex]\( MN \)[/tex].
Given:
[tex]\[ M = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \][/tex]
[tex]\[ MN = \begin{pmatrix} -6 & 3 \\ 4 & -2 \end{pmatrix} \][/tex]
We need to find matrix [tex]\( N \)[/tex] such that when [tex]\( M \)[/tex] is multiplied by [tex]\( N \)[/tex], the resulting matrix is [tex]\( MN \)[/tex].
### Step-by-Step Solution:
1. Express [tex]\( N \)[/tex] in a form that we can work with:
Since [tex]\( M \)[/tex] is a [tex]\( 2 \times 1 \)[/tex] column vector and [tex]\( MN \)[/tex] is a [tex]\( 2 \times 2 \)[/tex] matrix, [tex]\( N \)[/tex] must be a [tex]\( 1 \times 2 \)[/tex] row vector for the multiplication to be valid. Let's express [tex]\( N \)[/tex] as:
[tex]\[ N = \begin{pmatrix} a & b \end{pmatrix} \][/tex]
2. Set up the matrix multiplication [tex]\( M \times N \)[/tex]:
Multiply the column vector [tex]\( M \)[/tex] by the row vector [tex]\( N \)[/tex]:
[tex]\[ M \times N = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \begin{pmatrix} a & b \end{pmatrix} \][/tex]
This multiplication gives us:
[tex]\[ \begin{pmatrix} -3a & -3b \\ 2a & 2b \end{pmatrix} \][/tex]
3. Equate the result to the given [tex]\( MN \)[/tex]:
We are given that:
[tex]\[ \begin{pmatrix} -3a & -3b \\ 2a & 2b \end{pmatrix} = \begin{pmatrix} -6 & 3 \\ 4 & -2 \end{pmatrix} \][/tex]
From this equality, we can set up a system of equations:
- For the first column:
[tex]\[ -3a = -6 \][/tex]
[tex]\[ 2a = 4 \][/tex]
- For the second column:
[tex]\[ -3b = 3 \][/tex]
[tex]\[ 2b = -2 \][/tex]
4. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
From the first column, we solve:
[tex]\[ -3a = -6 \implies a = 2 \][/tex]
[tex]\[ 2a = 4 \implies a = 2 \][/tex]
Both equations confirm that [tex]\( a = 2 \)[/tex].
From the second column, we solve:
[tex]\[ -3b = 3 \implies b = -1 \][/tex]
[tex]\[ 2b = -2 \implies b = -1 \][/tex]
Both equations confirm that [tex]\( b = -1 \)[/tex].
### Conclusion:
The values [tex]\( a = 2 \)[/tex] and [tex]\( b = -1 \)[/tex] give us the matrix [tex]\( N \)[/tex]:
[tex]\[ N = \begin{pmatrix} 2 & -1 \end{pmatrix} \][/tex]
Thus, the matrix [tex]\( N \)[/tex] is:
[tex]\[ N = \begin{pmatrix} 2 & -1 \end{pmatrix} \][/tex]
Given:
[tex]\[ M = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \][/tex]
[tex]\[ MN = \begin{pmatrix} -6 & 3 \\ 4 & -2 \end{pmatrix} \][/tex]
We need to find matrix [tex]\( N \)[/tex] such that when [tex]\( M \)[/tex] is multiplied by [tex]\( N \)[/tex], the resulting matrix is [tex]\( MN \)[/tex].
### Step-by-Step Solution:
1. Express [tex]\( N \)[/tex] in a form that we can work with:
Since [tex]\( M \)[/tex] is a [tex]\( 2 \times 1 \)[/tex] column vector and [tex]\( MN \)[/tex] is a [tex]\( 2 \times 2 \)[/tex] matrix, [tex]\( N \)[/tex] must be a [tex]\( 1 \times 2 \)[/tex] row vector for the multiplication to be valid. Let's express [tex]\( N \)[/tex] as:
[tex]\[ N = \begin{pmatrix} a & b \end{pmatrix} \][/tex]
2. Set up the matrix multiplication [tex]\( M \times N \)[/tex]:
Multiply the column vector [tex]\( M \)[/tex] by the row vector [tex]\( N \)[/tex]:
[tex]\[ M \times N = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \begin{pmatrix} a & b \end{pmatrix} \][/tex]
This multiplication gives us:
[tex]\[ \begin{pmatrix} -3a & -3b \\ 2a & 2b \end{pmatrix} \][/tex]
3. Equate the result to the given [tex]\( MN \)[/tex]:
We are given that:
[tex]\[ \begin{pmatrix} -3a & -3b \\ 2a & 2b \end{pmatrix} = \begin{pmatrix} -6 & 3 \\ 4 & -2 \end{pmatrix} \][/tex]
From this equality, we can set up a system of equations:
- For the first column:
[tex]\[ -3a = -6 \][/tex]
[tex]\[ 2a = 4 \][/tex]
- For the second column:
[tex]\[ -3b = 3 \][/tex]
[tex]\[ 2b = -2 \][/tex]
4. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
From the first column, we solve:
[tex]\[ -3a = -6 \implies a = 2 \][/tex]
[tex]\[ 2a = 4 \implies a = 2 \][/tex]
Both equations confirm that [tex]\( a = 2 \)[/tex].
From the second column, we solve:
[tex]\[ -3b = 3 \implies b = -1 \][/tex]
[tex]\[ 2b = -2 \implies b = -1 \][/tex]
Both equations confirm that [tex]\( b = -1 \)[/tex].
### Conclusion:
The values [tex]\( a = 2 \)[/tex] and [tex]\( b = -1 \)[/tex] give us the matrix [tex]\( N \)[/tex]:
[tex]\[ N = \begin{pmatrix} 2 & -1 \end{pmatrix} \][/tex]
Thus, the matrix [tex]\( N \)[/tex] is:
[tex]\[ N = \begin{pmatrix} 2 & -1 \end{pmatrix} \][/tex]