Given `[(4,2),(-1,1)]`M = 6 I, where M is a matrix and I is the unit matrix or order `2xx2`.
(ii) Find the matrix M.

Answer

To solve the problem, we need to find the matrix \( M \) given the equation:

\[ \begin{pmatrix} 4 & 2 \\ -1 & 1 \end{pmatrix} M = 6I \]

where \( I \) is the identity matrix of order \( 2 \times 2 \).

### Step 1: Define the matrix \( M \)

Let the matrix \( M \) be represented as:

\[ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

### Step 2: Write the equation using the identity matrix

The identity matrix \( I \) of order \( 2 \times 2 \) is:

\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Thus, \( 6I \) is:

\[ 6I = 6 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} \]

### Step 3: Set up the matrix equation

Now we can rewrite the equation as:

\[ \begin{pmatrix} 4 & 2 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} \]

### Step 4: Perform the matrix multiplication

Carrying out the multiplication on the left side:

\[ \begin{pmatrix} 4a + 2c & 4b + 2d \\ -a + c & -b + d \end{pmatrix} = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} \]

### Step 5: Set up the system of equations

From the equality of the matrices, we can set up the following equations:

1. \( 4a + 2c = 6 \)
2. \( 4b + 2d = 0 \)
3. \( -a + c = 0 \)
4. \( -b + d = 6 \)

### Step 6: Solve the equations

From equation (3):

\[ c = a \]

Substituting \( c = a \) into equation (1):

\[ 4a + 2a = 6 \implies 6a = 6 \implies a = 1 \]

Now substituting \( a = 1 \) back into \( c \):

\[ c = 1 \]

Next, we substitute \( b \) into equation (2):

\[ 4b + 2d = 0 \implies 2d = -4b \implies d = -2b \]

Now substituting \( d = -2b \) into equation (4):

\[ -b - 2b = 6 \implies -3b = 6 \implies b = -2 \]

Now substituting \( b = -2 \) back into \( d \):

\[ d = -2(-2) = 4 \]

### Step 7: Write the final matrix \( M \)

Now we have:

\[ a = 1, \quad b = -2, \quad c = 1, \quad d = 4 \]

Thus, the matrix \( M \) is:

\[ M = \begin{pmatrix} 1 & -2 \\ 1 & 4 \end{pmatrix} \]

### Final Answer

\[ M = \begin{pmatrix} 1 & -2 \\ 1 & 4 \end{pmatrix} \]

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