Find the matrix X so that `X[[1 ,2 ,3], [4, 5 ,6]]=[[-7,-8,-9],[2,4,6]]`

Explanation

To find the matrix \( X \) such that

\[ X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}, \]

we will denote the matrix \( X \) as

\[ X = \begin{bmatrix} A & B \\ C & D \end{bmatrix}. \]

### Step 1: Set up the equation
We can express the multiplication of matrices as follows:

\[ \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} A + 4B & 2A + 5B & 3A + 6B \\ C + 4D & 2C + 5D & 3C + 6D \end{bmatrix}. \]

### Step 2: Equate the matrices
Now we equate the resulting matrix to the given matrix:

\[ \begin{bmatrix} A + 4B & 2A + 5B & 3A + 6B \\ C + 4D & 2C + 5D & 3C + 6D \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}. \]

From this, we can derive the following equations:

1. \( A + 4B = -7 \) (Equation 1)
2. \( 2A + 5B = -8 \) (Equation 2)
3. \( C + 4D = 2 \) (Equation 3)
4. \( 3C + 6D = 6 \) (Equation 4)

### Step 3: Solve for \( A \) and \( B \)
We will first solve for \( A \) and \( B \) using Equations 1 and 2.

From Equation 1, we can express \( A \) in terms of \( B \):

\[ A = -7 - 4B. \]

Substituting this expression for \( A \) into Equation 2:

\[ 2(-7 - 4B) + 5B = -8. \]

This simplifies to:

\[ -14 - 8B + 5B = -8 \implies -14 - 3B = -8 \implies -3B = 6 \implies B = -2. \]

Now substituting \( B = -2 \) back into Equation 1 to find \( A \):

\[ A + 4(-2) = -7 \implies A - 8 = -7 \implies A = 1. \]

### Step 4: Solve for \( C \) and \( D \)
Next, we solve for \( C \) and \( D \) using Equations 3 and 4.

From Equation 3, we can express \( C \) in terms of \( D \):

\[ C = 2 - 4D. \]

Substituting this expression for \( C \) into Equation 4:

\[ 3(2 - 4D) + 6D = 6. \]

This simplifies to:

\[ 6 - 12D + 6D = 6 \implies 6 - 6D = 6 \implies -6D = 0 \implies D = 0. \]

Now substituting \( D = 0 \) back into Equation 3 to find \( C \):

\[ C + 4(0) = 2 \implies C = 2. \]

### Step 5: Form the matrix \( X \)
Now we have all the values:

\[ A = 1, \quad B = -2, \quad C = 2, \quad D = 0. \]

Thus, the matrix \( X \) is:

\[ X = \begin{bmatrix} 1 & -2 \\ 2 & 0 \end{bmatrix}. \]

### Final Answer
The matrix \( X \) is

\[ \begin{bmatrix} 1 & -2 \\ 2 & 0 \end{bmatrix}. \]