Math Problem Statement
Is the following matrix invertible? (2, -3, 1; 4, 5, 6; 1, 0, 2)
Solution
To determine if the given matrix is invertible, we need to calculate its determinant. If the determinant is non-zero, the matrix is invertible. If the determinant is zero, the matrix is not invertible.
The matrix is:
2 & -3 & 1 \\ 4 & 5 & 6 \\ 1 & 0 & 2 \end{pmatrix}$$ The determinant of a 3x3 matrix $$A$$ can be calculated as: $$\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ Where: $$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ For matrix $$A$$, the elements are: $$a = 2, \, b = -3, \, c = 1, \, d = 4, \, e = 5, \, f = 6, \, g = 1, \, h = 0, \, i = 2$$ Let me calculate the determinant.The determinant of the given matrix is $$21$$, which is non-zero. Therefore, the matrix is invertible. Would you like any further details or have any questions? Here are five related questions you might find useful: 1. What does the determinant of a matrix represent geometrically? 2. How is the inverse of a 3x3 matrix calculated step by step? 3. How can we verify if a matrix is singular or not without computing the determinant? 4. What role does matrix inversion play in solving systems of linear equations? 5. How does row-reduction relate to determining whether a matrix is invertible? **Tip:** If a matrix has a non-zero determinant, it means the matrix represents a transformation that is one-to-one and onto in linear algebra.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Theory
Formulas
Determinant of a 3x3 matrix
Theorems
Invertibility Theorem
Determinant Theorem
Suitable Grade Level
College-level Linear Algebra