If for two matrices M and N, N =`[(3,2),(2,-1)]` and product `MxxN = [-1 4]` , find matrix M.

Answer

To find the matrix \( M \) given the matrices \( N \) and the product \( M \times N \), we will follow these steps:

### Step 1: Identify the matrices
We are given:
- Matrix \( N = \begin{pmatrix} 3 & 2 \\ 2 & -1 \end{pmatrix} \)
- Product \( M \times N = \begin{pmatrix} -1 & 4 \end{pmatrix} \)

### Step 2: Determine the order of matrix \( M \)
Since the product \( M \times N \) results in a \( 1 \times 2 \) matrix and \( N \) is a \( 2 \times 2 \) matrix, the matrix \( M \) must be of order \( 1 \times 2 \).

### Step 3: Assume the form of matrix \( M \)
Let matrix \( M \) be represented as:
\[ M = \begin{pmatrix} x & y \end{pmatrix} \]

### Step 4: Set up the matrix multiplication
The multiplication of \( M \) and \( N \) can be expressed as:
\[ M \times N = \begin{pmatrix} x & y \end{pmatrix} \times \begin{pmatrix} 3 & 2 \\ 2 & -1 \end{pmatrix} \]
Calculating this gives:
\[ M \times N = \begin{pmatrix} 3x + 2y & 2x - y \end{pmatrix} \]

### Step 5: Set up equations from the product
We know that:
\[ M \times N = \begin{pmatrix} -1 & 4 \end{pmatrix} \]
This leads to the equations:
1. \( 3x + 2y = -1 \)
2. \( 2x - y = 4 \)

### Step 6: Solve the equations
We will solve these equations simultaneously.

From the second equation:
\[ 2x - y = 4 \implies y = 2x - 4 \]

Substituting \( y \) in the first equation:
\[ 3x + 2(2x - 4) = -1 \]
This simplifies to:
\[ 3x + 4x - 8 = -1 \]
Combining like terms:
\[ 7x - 8 = -1 \]
Adding 8 to both sides:
\[ 7x = 7 \implies x = 1 \]

Now substituting \( x = 1 \) back into the equation for \( y \):
\[ y = 2(1) - 4 = 2 - 4 = -2 \]

### Step 7: Write the final matrix \( M \)
Thus, the matrix \( M \) is:
\[ M = \begin{pmatrix} 1 & -2 \end{pmatrix} \]

### Final Answer
The matrix \( M \) is:
\[ \begin{pmatrix} 1 & -2 \end{pmatrix} \]