`[(1,3),(-2,4)]+[(11,5),(-6,12)]=K[(3,2),(J,M)]`. Find the value of K+J+M.

Options

A. 2
B. 4
C. 6
D. 7

Explanation

To solve the equation \([(1,3),(-2,4)] + [(11,5),(-6,12)] = K[(3,2),(J,M)]\), we will follow these steps:

### Step 1: Add the two matrices on the left side.
We will add the corresponding entries of the two matrices.

\[ \begin{pmatrix} 1 & 3 \\ -2 & 4 \end{pmatrix} + \begin{pmatrix} 11 & 5 \\ -6 & 12 \end{pmatrix} = \begin{pmatrix} 1 + 11 & 3 + 5 \\ -2 + (-6) & 4 + 12 \end{pmatrix} \]

Calculating each entry:

- First entry: \(1 + 11 = 12\)
- Second entry: \(3 + 5 = 8\)
- Third entry: \(-2 + (-6) = -8\)
- Fourth entry: \(4 + 12 = 16\)

So, the resulting matrix is:

\[ \begin{pmatrix} 12 & 8 \\ -8 & 16 \end{pmatrix} \]

### Step 2: Set the resulting matrix equal to \(K[(3,2),(J,M)]\).
This gives us:

\[ \begin{pmatrix} 12 & 8 \\ -8 & 16 \end{pmatrix} = K \begin{pmatrix} 3 & 2 \\ J & M \end{pmatrix} \]

### Step 3: Multiply the right side by \(K\).
This results in:

\[ \begin{pmatrix} K \cdot 3 & K \cdot 2 \\ K \cdot J & K \cdot M \end{pmatrix} \]

### Step 4: Equate the corresponding entries.
Now we equate the corresponding entries from both matrices:

1. \(K \cdot 3 = 12\)
2. \(K \cdot 2 = 8\)
3. \(K \cdot J = -8\)
4. \(K \cdot M = 16\)

### Step 5: Solve for \(K\).
From the first equation:

\[ K \cdot 3 = 12 \implies K = \frac{12}{3} = 4 \]

### Step 6: Verify \(K\) using the second equation.
Using \(K = 4\) in the second equation:

\[ K \cdot 2 = 8 \implies 4 \cdot 2 = 8 \quad \text{(True)} \]

### Step 7: Find \(J\) and \(M\).
Using \(K = 4\):

1. From \(K \cdot J = -8\):
\[ 4 \cdot J = -8 \implies J = \frac{-8}{4} = -2 \]

2. From \(K \cdot M = 16\):
\[ 4 \cdot M = 16 \implies M = \frac{16}{4} = 4 \]

### Step 8: Calculate \(K + J + M\).
Now we can calculate:

\[ K + J + M = 4 + (-2) + 4 = 6 \]

Thus, the final answer is:

\[ \boxed{6} \]