If `A=[a b c d]` (where `b c!=0` ) satisfies the equations `x^2+k=0,t h e n` `a+d=0` b. `K=-|A|` c. `k=|A|` d. none of these

Options

A. `a+d=0`
B. `k=-|A|`
C. `k=|A|`
D. none of these

Explanation

To solve the problem, we need to analyze the given matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) and the equation \( x^2 + k = 0 \). We will derive the relationships step by step.

### Step 1: Understand the Equation
The equation \( x^2 + k = 0 \) implies that if we substitute \( A \) for \( x \), we get:
\[ A^2 + kI = 0 \]
where \( I \) is the identity matrix.

### Step 2: Calculate \( A^2 \)
To find \( A^2 \), we perform the matrix multiplication:
\[ A^2 = A \cdot A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
Calculating this gives:
\[ A^2 = \begin{bmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{bmatrix} \]

### Step 3: Substitute into the Equation
Now, substituting \( A^2 \) into the equation \( A^2 + kI = 0 \):
\[ \begin{bmatrix} a^2 + bc + k & ab + bd \\ ac + cd & bc + d^2 + k \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]
This gives us two equations:
1. \( a^2 + bc + k = 0 \)
2. \( bc + d^2 + k = 0 \)

### Step 4: Equate the Two Equations
From the two equations, we can express \( k \):
1. From the first equation: \( k = - (a^2 + bc) \)
2. From the second equation: \( k = - (bc + d^2) \)

Setting these equal to each other:
\[ -(a^2 + bc) = -(bc + d^2) \]
This simplifies to:
\[ a^2 = d^2 \]

### Step 5: Analyze the Result
Since \( a^2 = d^2 \), we can conclude:
\[ a = d \quad \text{or} \quad a = -d \]
Given that \( b \) and \( c \) are not equal to zero, we focus on the case \( a + d = 0 \) (i.e., \( a = -d \)).

### Step 6: Find the Determinant
The determinant of matrix \( A \) is given by:
\[ |A| = ad - bc \]
Substituting \( d = -a \) into the determinant:
\[ |A| = a(-a) - bc = -a^2 - bc \]

### Step 7: Find the Value of \( k \)
From the first equation, we substitute \( k \):
\[ k = - (a^2 + bc) \]
Thus, we have:
\[ k = -|A| \]

### Conclusion
From the analysis, we conclude:
1. \( a + d = 0 \)
2. \( k = -|A| \)

### Final Answer
The correct options are:
- a. \( a + d = 0 \)
- b. \( k = -|A| \)