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 and
Given:
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}$$ First, let's compute $$2A$$ and $$3B$$: $$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}$$ $$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}$$ ### Step 2: Compute $$2A + 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}$$ ### Step 3: Use the condition of singularity A matrix is singular if its determinant is zero. So, we need to find the determinant of the matrix $$2A + 3B$$ and set it equal to zero. Let’s calculate the determinant: $$\text{det}(2A + 3B) = \begin{vmatrix} 11 & 12 & 2\lambda + 16 \\ 13 & 14 & 31 \\ 12 & 13 & 32 \end{vmatrix}$$ Using the formula for the determinant of a 3x3 matrix: $$\text{det}(M) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ Where the matrix $$M$$ is: $$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ Substituting in the values: $$\text{det}(2A + 3B) = 11\left(14 \times 32 - 31 \times 13\right) - 12\left(13 \times 32 - 31 \times 12\right) + \left(2\lambda + 16\right)\left(13 \times 13 - 14 \times 12\right)$$ Calculate each of these: - $$14 \times 32 - 31 \times 13 = 448 - 403 = 45$$ - $$13 \times 32 - 31 \times 12 = 416 - 372 = 44$$ - $$13 \times 13 - 14 \times 12 = 169 - 168 = 1$$ So: $$\text{det}(2A + 3B) = 11(45) - 12(44) + (2\lambda + 16)(1)$$ $$\text{det}(2A + 3B) = 495 - 528 + 2\lambda + 16$$ $$\text{det}(2A + 3B) = 2\lambda - 17$$ For the matrix to be singular: $$2\lambda - 17 = 0$$ ### Step 4: Solve for $$\lambda$$ $$2\lambda = 17$$ Thus, the value of $$2\lambda$$ is $$\boxed{17}$$. Would you like any more details or have any questions? Here are 5 related questions: 1. What properties of matrices can make a matrix singular? 2. How do you determine the determinant of a 3x3 matrix? 3. What are the implications of a matrix being singular? 4. Can a non-square matrix be singular? 5. How does the determinant relate to the invertibility of a matrix? **Tip:** When calculating determinants, always simplify as much as possible before final calculations to reduce potential errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrices
Determinants
Singularity
Formulas
Determinant of a 3x3 matrix
Theorems
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Suitable Grade Level
Advanced High School