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

Options

A. 1
B. 0
C. 10
D. 5

Explanation

To solve the problem step by step, we start with the given matrices and the equations derived from them.

### Step 1: Write down the matrices
We have:
- Matrix \( A = \begin{pmatrix} 0 & 2 \\ 3 & -4 \end{pmatrix} \)
- Matrix \( kA = \begin{pmatrix} 0 & 3a \\ 2b & 24 \end{pmatrix} \)

### Step 2: Express the matrix \( kA \)
The matrix \( kA \) can be expressed as:
\[ kA = k \begin{pmatrix} 0 & 2 \\ 3 & -4 \end{pmatrix} = \begin{pmatrix} 0 & 2k \\ 3k & -4k \end{pmatrix} \]

### Step 3: Set the matrices equal to each other
Now, we equate the two expressions for \( kA \):
\[ \begin{pmatrix} 0 & 2k \\ 3k & -4k \end{pmatrix} = \begin{pmatrix} 0 & 3a \\ 2b & 24 \end{pmatrix} \]

### Step 4: Set up equations from the matrix equality
From the equality of the matrices, we can derive the following equations:
1. \( 2k = 3a \)
2. \( 3k = 2b \)
3. \( -4k = 24 \)

### Step 5: Solve for \( k \)
From the third equation:
\[ -4k = 24 \implies k = -6 \]

### Step 6: Substitute \( k \) into the first two equations
Now substitute \( k = -6 \) into the first equation:
\[ 2(-6) = 3a \implies -12 = 3a \implies a = -4 \]

Now substitute \( k = -6 \) into the second equation:
\[ 3(-6) = 2b \implies -18 = 2b \implies b = -9 \]

### Step 7: Calculate \( b - a - k \)
Now, we need to find \( b - a - k \):
\[ b - a - k = -9 - (-4) - (-6) = -9 + 4 + 6 = -9 + 10 = 1 \]

### Final Answer
Thus, the value of \( b - a - k \) is \( \boxed{1} \).

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