If A = `[{:(0,3),(4,5):}]` and kA = `[{:(0, 4a),(3b, 60):}]` , then value of k , a and b are respectively

Options

A. 12,19,16
B. 9, 12 , 16
C. 12, 9, 16
D. 16, 9, 12

Answer

To solve the problem, we need to find the values of \( k \), \( a \), and \( b \) given the matrices \( A \) and \( kA \).

### Given:
- \( A = \begin{pmatrix} 0 & 3 \\ 4 & 5 \end{pmatrix} \)
- \( kA = \begin{pmatrix} 0 & 4a \\ 3b & 60 \end{pmatrix} \)

### Step 1: Understanding the Matrix Multiplication
When we multiply a matrix \( A \) by a scalar \( k \), each element of the matrix is multiplied by \( k \). Therefore, we can express \( kA \) as follows:
\[ kA = k \begin{pmatrix} 0 & 3 \\ 4 & 5 \end{pmatrix} = \begin{pmatrix} k \cdot 0 & k \cdot 3 \\ k \cdot 4 & k \cdot 5 \end{pmatrix} = \begin{pmatrix} 0 & 3k \\ 4k & 5k \end{pmatrix} \]

### Step 2: Setting Up the Equations
Now, we can equate the corresponding elements of \( kA \) and the given matrix \( kA \):
1. From the first element: \( k \cdot 0 = 0 \) (which is always true)
2. From the second element: \( 3k = 4a \)
3. From the third element: \( 4k = 3b \)
4. From the fourth element: \( 5k = 60 \)

### Step 3: Solving for \( k \)
From the equation \( 5k = 60 \):
\[ k = \frac{60}{5} = 12 \]

### Step 4: Substituting \( k \) to Find \( a \)
Now, substitute \( k = 12 \) into the equation \( 3k = 4a \):
\[ 3(12) = 4a \implies 36 = 4a \implies a = \frac{36}{4} = 9 \]

### Step 5: Substituting \( k \) to Find \( b \)
Next, substitute \( k = 12 \) into the equation \( 4k = 3b \):
\[ 4(12) = 3b \implies 48 = 3b \implies b = \frac{48}{3} = 16 \]

### Final Values
Thus, the values of \( k \), \( a \), and \( b \) are:
- \( k = 12 \)
- \( a = 9 \)
- \( b = 16 \)

### Conclusion
The values of \( k \), \( a \), and \( b \) respectively are \( 12, 9, 16 \).

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