Let A and B are two matrices of same order i.e.
where `A=[(1,-3,2),(2,K,5),(4,2,1)], B=[(2,1,3),(4,2,4),(3,3,5)]`
If A is singular matrix then tr(A + B) is equal to

Options

A. 5
B. 3
C. 4
D. 6

Answer

To solve the problem, we need to find the trace of the sum of matrices A and B, given that matrix A is singular. Here’s a step-by-step solution:

### Step 1: Define the matrices A and B
Given:
\[ A = \begin{pmatrix} 1 & -3 & 2 \\ 2 & K & 5 \\ 4 & 2 & 1 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 & 1 & 3 \\ 4 & 2 & 4 \\ 3 & 3 & 5 \end{pmatrix} \]

### Step 2: Understand the condition of A being singular
A matrix is singular if its determinant is zero. Therefore, we need to find the determinant of matrix A and set it to zero.

### Step 3: Calculate the determinant of A
The determinant of a 3x3 matrix can be calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For matrix A:
\[ \text{det}(A) = 1 \cdot (K \cdot 1 - 5 \cdot 2) - (-3) \cdot (2 \cdot 1 - 5 \cdot 4) + 2 \cdot (2 \cdot 2 - K \cdot 4) \]
Calculating each term:
1. \( 1 \cdot (K - 10) = K - 10 \)
2. \( -(-3) \cdot (2 - 20) = 3 \cdot (-18) = -54 \)
3. \( 2 \cdot (4 - 4K) = 8 - 8K \)

Combining these:
\[ \text{det}(A) = K - 10 - 54 + 8 - 8K = -7K - 56 \]

### Step 4: Set the determinant to zero
Since A is singular:
\[ -7K - 56 = 0 \]
Solving for K:
\[ -7K = 56 \implies K = -8 \]

### Step 5: Find the trace of A
The trace of a matrix is the sum of its diagonal elements. For matrix A:
\[ \text{tr}(A) = 1 + K + 1 = 1 - 8 + 1 = -6 \]

### Step 6: Find the trace of B
For matrix B:
\[ \text{tr}(B) = 2 + 2 + 5 = 9 \]

### Step 7: Calculate the trace of A + B
The trace of the sum of two matrices is the sum of their traces:
\[ \text{tr}(A + B) = \text{tr}(A) + \text{tr}(B) = -6 + 9 = 3 \]

### Final Answer
The trace of \( A + B \) is equal to \( 3 \).

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