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 determine the value of given that the matrix is singular.
Step 1: Compute the matrix
Given the matrix :
1 & 3 & \lambda + 2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{pmatrix}$$ Now, calculate $$2A$$: $$2A = 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}$$ ### Step 2: Compute the matrix $$3B$$ Given the matrix $$B$$: $$B = \begin{pmatrix} 3 & 2 & 4 \\ 3 & 2 & 5 \\ 2 & 1 & 4 \end{pmatrix}$$ Now, calculate $$3B$$: $$3B = 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}$$ ### Step 3: Compute the matrix $$2A + 3B$$ Add the matrices $$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}$$ ### Step 4: Determine when $$2A + 3B$$ is singular A matrix is singular if its determinant is zero. Therefore, calculate the determinant of the matrix $$2A + 3B$$: $$\text{det}(2A + 3B) = \begin{vmatrix} 11 & 12 & 2\lambda + 16 \\ 13 & 14 & 31 \\ 12 & 13 & 32 \end{vmatrix}$$ Using the determinant formula for a 3x3 matrix: $$\text{det}(2A + 3B) = 11\begin{vmatrix}14 & 31 \\ 13 & 32\end{vmatrix} - 12\begin{vmatrix}13 & 31 \\ 12 & 32\end{vmatrix} + (2\lambda + 16)\begin{vmatrix}13 & 14 \\ 12 & 13\end{vmatrix}$$ Now, let's compute each of these smaller 2x2 determinants: $$\begin{vmatrix}14 & 31 \\ 13 & 32\end{vmatrix} = 14(32) - 31(13) = 448 - 403 = 45$$ $$\begin{vmatrix}13 & 31 \\ 12 & 32\end{vmatrix} = 13(32) - 31(12) = 416 - 372 = 44$$ $$\begin{vmatrix}13 & 14 \\ 12 & 13\end{vmatrix} = 13(13) - 14(12) = 169 - 168 = 1$$ So, substituting back: $$\text{det}(2A + 3B) = 11(45) - 12(44) + (2\lambda + 16)(1)$$ $$\text{det}(2A + 3B) = 495 - 528 + 2\lambda + 16 = 2\lambda - 17$$ ### Step 5: Set determinant to zero and solve for $$\lambda$$ Since the matrix is singular, set the determinant equal to zero: $$2\lambda - 17 = 0$$ $$2\lambda = 17$$ Thus, $$2\lambda = 17$$. Would you like more details or have any other questions? ### Related Questions: 1. How do you find the determinant of a 3x3 matrix? 2. What is the significance of a singular matrix in linear algebra? 3. How do matrix operations like addition and scalar multiplication affect the determinant? 4. Can you explain the concept of linear independence in the context of matrices? 5. What is the geometric interpretation of a singular matrix? ### Tip: When checking if a matrix is singular, always remember that a singular matrix has a determinant of zero, indicating that its rows or columns are linearly dependent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Determinants
Linear Algebra
Formulas
Determinant of a 3x3 matrix
Theorems
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Suitable Grade Level
Advanced High School