Math Problem Statement
Solution
To determine if a matrix is invertible, we need to calculate its determinant. If the determinant of a matrix is non-zero, then the matrix is invertible. For the given matrix:
we'll calculate the determinant by expanding along the first row.
Step-by-Step Determinant Calculation
The determinant of a matrix is calculated as:
For our matrix:
- , ,
- , ,
- , ,
Substitute these values into the determinant formula:
Calculate Each Term
-
For the first term:
- So,
-
For the second term:
- So,
-
For the third term:
- So,
Combine the Terms
Since the determinant is , which is not zero, the matrix is invertible.
Would you like further details on any steps, or do you have any questions?
Related Questions
- What does it mean if a matrix has a determinant of zero?
- How do you calculate the inverse of a matrix?
- Can a non-square matrix be invertible?
- How does the determinant change if we swap two rows in a matrix?
- What is the geometric interpretation of a matrix's determinant?
Tip
For larger matrices, using row or column expansion with the most zeros simplifies determinant calculations significantly.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrix Inversion
Formulas
Determinant of a 3x3 matrix: det = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
Invertibility of a Matrix (A matrix is invertible if its determinant is non-zero)
Suitable Grade Level
Grades 11-12