If `[(2,-3),(6,5)][(1,0),(2,3)]=[(-4,-9),(16,15)]`
Write the equation after applying elementary column transformation `C_(2)rarrC_(2)+2C_(1)`

Answer

To solve the problem, we need to apply the elementary column transformation \( C_2 \rightarrow C_2 + 2C_1 \) to the given matrices. We will follow these steps:

### Step 1: Write down the original matrices
We start with the original matrices:
\[ A = \begin{pmatrix} 2 & -3 \\ 6 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}, \quad C = \begin{pmatrix} -4 & -9 \\ 16 & 15 \end{pmatrix} \]

### Step 2: Apply the column transformation
We need to apply the transformation \( C_2 \rightarrow C_2 + 2C_1 \) to matrix \( B \).

The first column \( C_1 \) of matrix \( B \) is:
\[ C_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \]

The second column \( C_2 \) of matrix \( B \) is:
\[ C_2 = \begin{pmatrix} 0 \\ 3 \end{pmatrix} \]

Now, we compute the new second column:
\[ C_2' = C_2 + 2C_1 = \begin{pmatrix} 0 \\ 3 \end{pmatrix} + 2 \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 0 + 2 \cdot 1 \\ 3 + 2 \cdot 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 7 \end{pmatrix} \]

### Step 3: Form the new matrix after transformation
After applying the transformation, the new matrix \( B' \) becomes:
\[ B' = \begin{pmatrix} 1 & 2 \\ 2 & 7 \end{pmatrix} \]

### Step 4: Write the new equation
Now we write the equation after applying the transformation:
\[ \begin{pmatrix} 2 & -3 \\ 6 & 5 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & 7 \end{pmatrix} = \begin{pmatrix} -4 & -9 \\ 16 & 15 \end{pmatrix} \]

### Final Result
Thus, the equation after applying the elementary column transformation is:
\[ \begin{pmatrix} 2 & -3 \\ 6 & 5 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & 7 \end{pmatrix} = \begin{pmatrix} -4 & -9 \\ 16 & 15 \end{pmatrix} \]

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