Math Problem Statement
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?
Solution
To solve the problem, we need to find the value of given that the matrix is singular. Here's the step-by-step solution:
Step 1: Calculate and
Given:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Determinants
Singular Matrices
Formulas
Determinant calculation using cofactor expansion
Theorems
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Suitable Grade Level
Advanced High School or College