Math Problem Statement
For what value of a is the following matrix singular? (2a −1 −8 3)
Solution
A matrix is singular if its determinant is zero. For the given matrix:
2a - 1 & 1 \\ -8 & 3 \end{pmatrix}$$ we calculate the determinant using the formula: $$\text{det}(A) = (2a - 1)(3) - (-8)(1)$$ Simplifying the determinant expression: $$\text{det}(A) = 3(2a - 1) + 8$$ Expanding: $$\text{det}(A) = 6a - 3 + 8$$ $$\text{det}(A) = 6a + 5$$ To find when the matrix is singular, set the determinant equal to zero: $$6a + 5 = 0$$ Solving for $$a$$: $$6a = -5$$ $$a = -\frac{5}{6}$$ Thus, the matrix is singular when $$a = -\frac{5}{6}$$. Do you have any questions, or would you like further details? Here are 5 related questions to expand this topic: 1. How do you find the determinant of a 3x3 matrix? 2. What does it mean for a matrix to be non-singular? 3. Can a matrix have more than one value of $$a$$ that makes it singular? 4. How does the determinant relate to the inverse of a matrix? 5. What are some applications of singular matrices in real-world problems? **Tip:** The determinant of a matrix provides valuable information about its invertibility. If the determinant is zero, the matrix does not have an inverse.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Determinants
Formulas
det(A) = ad - bc for a 2x2 matrix [[a, b], [c, d]]
Theorems
A matrix is singular if and only if its determinant is zero
Suitable Grade Level
Grades 11-12 or early college