2.Let A and B be two matrices of same order 3 x 3 where A=[(1,3, λ+2),(2,4,8),(3,5,10)] B=[(3,2,4),(3,2,5),(2,1,4)]. If matrix (2A+3B) is singular then find the value of 2λ?

To solve the problem, we need to find the value of \( 2\lambda \) given that the matrix \( (2A + 3B) \) is singular. Here's the step-by-step solution:

### Step 1: Calculate \( 2A \) and \( 3B \)
Given:
\[
A = \begin{pmatrix} 1 & 3 & \lambda + 2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{pmatrix}, \quad
B = \begin{pmatrix} 3 & 2 & 4 \\ 3 & 2 & 5 \\ 2 & 1 & 4 \end{pmatrix}
\]

First, calculate \( 2A \):
\[
2A = 2 \times A = 2 \times \begin{pmatrix} 1 & 3 & \lambda + 2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{pmatrix} = \begin{pmatrix} 2 & 6 & 2\lambda + 4 \\ 4 & 8 & 16 \\ 6 & 10 & 20 \end{pmatrix}
\]

Next, calculate \( 3B \):
\[
3B = 3 \times B = 3 \times \begin{pmatrix} 3 & 2 & 4 \\ 3 & 2 & 5 \\ 2 & 1 & 4 \end{pmatrix} = \begin{pmatrix} 9 & 6 & 12 \\ 9 & 6 & 15 \\ 6 & 3 & 12 \end{pmatrix}
\]

### Step 2: Calculate \( 2A + 3B \)
Now, add the matrices \( 2A \) and \( 3B \):
\[
2A + 3B = \begin{pmatrix} 2 & 6 & 2\lambda + 4 \\ 4 & 8 & 16 \\ 6 & 10 & 20 \end{pmatrix} + \begin{pmatrix} 9 & 6 & 12 \\ 9 & 6 & 15 \\ 6 & 3 & 12 \end{pmatrix} = \begin{pmatrix} 11 & 12 & 2\lambda + 16 \\ 13 & 14 & 31 \\ 12 & 13 & 32 \end{pmatrix}
\]

### Step 3: Determine the determinant of \( 2A + 3B \)
Since \( 2A + 3B \) is singular, its determinant must be zero:
\[
\text{Determinant of } (2A + 3B) = \begin{vmatrix} 11 & 12 & 2\lambda + 16 \\ 13 & 14 & 31 \\ 12 & 13 & 32 \end{vmatrix} = 0
\]

Calculate the determinant using cofactor expansion:
\[
\text{Determinant} = 11 \times \begin{vmatrix} 14 & 31 \\ 13 & 32 \end{vmatrix} - 12 \times \begin{vmatrix} 13 & 31 \\ 12 & 32 \end{vmatrix} + (2\lambda + 16) \times \begin{vmatrix} 13 & 14 \\ 12 & 13 \end{vmatrix}
\]

First, calculate the minor determinants:
\[
\begin{vmatrix} 14 & 31 \\ 13 & 32 \end{vmatrix} = (14 \times 32) - (31 \times 13) = 448 - 403 = 45
\]
\[
\begin{vmatrix} 13 & 31 \\ 12 & 32 \end{vmatrix} = (13 \times 32) - (31 \times 12) = 416 - 372 = 44
\]
\[
\begin{vmatrix} 13 & 14 \\ 12 & 13 \end{vmatrix} = (13 \times 13) - (14 \times 12) = 169 - 168 = 1
\]

Now substitute back into the determinant expression:
\[
\text{Determinant} = 11 \times 45 - 12 \times 44 + (2\lambda + 16) \times 1 = 495 - 528 + 2\lambda + 16 = 2\lambda - 17
\]

### Step 4: Solve for \( \lambda \)
Since the determinant must be zero for the matrix to be singular:
\[
2\lambda - 17 = 0
\]
\[
2\lambda = 17
\]

Thus, the value of \( 2\lambda \) is \( \boxed{17} \).

Would you like more details on any of the steps? 

Here are some related questions for further exploration:
1. How do you determine if a matrix is singular?
2. What is the cofactor expansion method for calculating determinants?
3. Can you generalize the determinant calculation for \( n \times n \) matrices?
4. What are some properties of determinants that can simplify calculations?
5. How does the determinant relate to the invertibility of a matrix?

**Tip:** Always check the determinant value when determining whether a matrix is singular—it must be zero for the matrix to be singular.

Learn how to find the value of 2λ for the singular matrix (2A + 3B), where A and B are 3x3 matrices. This problem involves matrix algebra, determinant calculations, and singular matrices. Suitable for advanced high school or college-level students.

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