If `X+Y=[(5,2),(0,9)] and X-Y=[(3, 6), (0,-1)] and X=[(k,k),(0,k)]`, then k =

Options

A. 8
B. 2
C. 4
D. 1

Answer

To solve the problem, we have the following equations:

1. \( X + Y = \begin{pmatrix} 5 & 2 \\ 0 & 9 \end{pmatrix} \)
2. \( X - Y = \begin{pmatrix} 3 & 6 \\ 0 & -1 \end{pmatrix} \)
3. \( X = \begin{pmatrix} k & k \\ 0 & k \end{pmatrix} \)

We need to find the value of \( k \).

### Step-by-step Solution:

**Step 1: Add the two equations**
We start by adding the two equations \( X + Y \) and \( X - Y \):

\[ (X + Y) + (X - Y) = \begin{pmatrix} 5 & 2 \\ 0 & 9 \end{pmatrix} + \begin{pmatrix} 3 & 6 \\ 0 & -1 \end{pmatrix} \]

This simplifies to:

\[ 2X = \begin{pmatrix} 5 + 3 & 2 + 6 \\ 0 + 0 & 9 - 1 \end{pmatrix} = \begin{pmatrix} 8 & 8 \\ 0 & 8 \end{pmatrix} \]

**Step 2: Solve for \( X \)**
Now, divide both sides by 2 to solve for \( X \):

\[ X = \frac{1}{2} \begin{pmatrix} 8 & 8 \\ 0 & 8 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 0 & 4 \end{pmatrix} \]

**Step 3: Compare with the given form of \( X \)**
We know that \( X = \begin{pmatrix} k & k \\ 0 & k \end{pmatrix} \). Now we can compare this with our calculated value of \( X \):

\[ \begin{pmatrix} k & k \\ 0 & k \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 0 & 4 \end{pmatrix} \]

From this comparison, we can see that \( k = 4 \).

### Final Answer:
Thus, the value of \( k \) is \( 4 \).

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