If `A={:[(0,2),(3,-4)]:}and kA={:[(0,3a),(2b,24)]:}`, then the values of k,a,b are respectively.

Options

A. `-6,-12,-18`
B. `-6,4,9`
C. `-6,-4,-9`
D. `-6,12,18`

Explanation

To solve the problem, we start with the given matrices:

1. **Matrix A**:
\[ A = \begin{pmatrix} 0 & 2 \\ 3 & -4 \end{pmatrix} \]

2. **Matrix \( kA \)**:
\[ kA = \begin{pmatrix} 0 & 3a \\ 2b & 24 \end{pmatrix} \]

### Step 1: Express \( kA \) in terms of \( k \) and \( A \)

When we multiply matrix \( A \) by \( k \), we get:
\[ kA = k \begin{pmatrix} 0 & 2 \\ 3 & -4 \end{pmatrix} = \begin{pmatrix} 0 & 2k \\ 3k & -4k \end{pmatrix} \]

### Step 2: Set up equations by equating corresponding elements

Now, we equate the corresponding elements of \( kA \) and the given matrix \( kA \):
1. From the first element (top left):
\[ 0 = 0 \quad \text{(This is always true)} \]

2. From the second element (top right):
\[ 2k = 3a \quad \text{(Equation 1)} \]

3. From the third element (bottom left):
\[ 3k = 2b \quad \text{(Equation 2)} \]

4. From the fourth element (bottom right):
\[ -4k = 24 \quad \text{(Equation 3)} \]

### Step 3: Solve for \( k \) using Equation 3

From Equation 3:
\[ -4k = 24 \]
Dividing both sides by -4:
\[ k = -6 \]

### Step 4: Substitute \( k \) into Equations 1 and 2 to find \( a \) and \( b \)

**Finding \( a \)**:
Substituting \( k = -6 \) into Equation 1:
\[ 2(-6) = 3a \implies -12 = 3a \]
Dividing both sides by 3:
\[ a = -4 \]

**Finding \( b \)**:
Substituting \( k = -6 \) into Equation 2:
\[ 3(-6) = 2b \implies -18 = 2b \]
Dividing both sides by 2:
\[ b = -9 \]

### Final Values
Thus, the values of \( k, a, b \) are:
\[ k = -6, \quad a = -4, \quad b = -9 \]

### Summary
The values of \( k, a, b \) are respectively:
\[ \text{Answer: } k = -6, a = -4, b = -9 \]