To solve for the values of $k$ and $m$, we need to perform matrix multiplication and equate the resulting matrix to the given result matrix.

We are given: $A = \begin{bmatrix} 2 & -2 \\ 4 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & m \\ k & 2 \end{bmatrix}, \quad AB = \begin{bmatrix} -4 & 4 \\ 7 & 18 \end{bmatrix}$

Step 1: Calculate $AB$ in terms of $k$ and $m$

The multiplication of matrices $A$ and $B$ (resulting in a $2 \times 2$ matrix) is as follows:

$AB = \begin{bmatrix} 2 & -2 \\ 4 & 1 \end{bmatrix} \begin{bmatrix} 1 & m \\ k & 2 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 + (-2) \cdot k & 2 \cdot m + (-2) \cdot 2 \\ 4 \cdot 1 + 1 \cdot k & 4 \cdot m + 1 \cdot 2 \end{bmatrix}$

Simplifying each entry:

1. Top left: $2 \cdot 1 + (-2) \cdot k = 2 - 2k$
2. Top right: $2 \cdot m + (-2) \cdot 2 = 2m - 4$
3. Bottom left: $4 \cdot 1 + 1 \cdot k = 4 + k$
4. Bottom right: $4 \cdot m + 1 \cdot 2 = 4m + 2$

So, $AB = \begin{bmatrix} 2 - 2k & 2m - 4 \\ 4 + k & 4m + 2 \end{bmatrix}$

Step 2: Set up equations by equating $AB$ to the given matrix

Since $AB = \begin{bmatrix} -4 & 4 \\ 7 & 18 \end{bmatrix}$, we can set up equations by equating corresponding elements:

1. $2 - 2k = -4$
2. $2m - 4 = 4$
3. $4 + k = 7$
4. $4m + 2 = 18$

Step 3: Solve the equations

Solving for $k$:

From equation 1: $2 - 2k = -4 \Rightarrow -2k = -6 \Rightarrow k = 3$

Solving for $m$:

From equation 2: $2m - 4 = 4 \Rightarrow 2m = 8 \Rightarrow m = 4$

Step 4: Verify the values in the remaining equations

1. Substituting $k = 3$ in equation 3: $4 + k = 4 + 3 = 7 \quad \text{(True)}$

2. Substituting $m = 4$ in equation 4: $4m + 2 = 4 \cdot 4 + 2 = 16 + 2 = 18 \quad \text{(True)}$

Conclusion

The values of $k$ and $m$ that satisfy the matrix equation are: $k = 3, \quad m = 4$

Would you like further details or have any questions?

Here are some related questions you may find helpful:

1. How do you perform matrix multiplication for different-sized matrices?
2. What are the properties of matrix multiplication?
3. How can matrices be used to solve systems of linear equations?
4. What is the significance of the determinant in matrix operations?
5. How can you verify if two matrices are inverses of each other?

Tip: When solving matrix equations, double-check each element by substitution to avoid calculation errors.