Determinant Algebra, Cramer's Rule & Inverses
“Axiomatic multilinearity, geometric volume interpretation, determinant identities, and adjugate inverses.”
1.Determinants: Geometric Meaning & Algebraic Properties
§ 2.1.The Determinant is a scalar value associated with any square matrix n x n representing the signed scaling factor of n-dimensional volume.
Key Axiomatic Properties:
1. det(I) = 1.
2. Swapping two rows multiplies determinant by -1.
3. If two rows/columns are identical or linearly dependent, det(A) = 0.
4. Adding a scalar multiple of one row to another preserves det(A).
5. For triangular or diagonal matrices, det(A) = prod_{i=1}^n a_{ii}.
The Adjugate & Matrix Inverse:
The classical adjoint adj(A) = C^T, where C is the cofactor matrix.
A adj(A) = det(A) I.
A^{-1} = (1 / det(A)) * adj(A).
2. Worked Practice Problem: Compound Determinant Identity
Carefully factor the scalar 2 out of the 3x3 matrix: factor is 2^3 = 8.
Problem: Let A be an invertible 3 x 3 real matrix such that det(A) = 5. Find the value of: det( 2 A^{-1} A^T * adj(A) ).
Step 1: Identify matrix dimensions and scalar scaling:
Matrix dimension n = 3.
For any scalar c and 3 x 3 matrix M, det(c M) = c^3 det(M).
Here c = 2, so the scalar contributes factor 2^3 = 8.
Step 2: Apply product rule det(M1 M2 M3) = det(M1) det(M2) det(M3):
det( 2 A^{-1} A^T adj(A) ) = 8 det(A^{-1}) det(A^T) det(adj(A)).
Step 3: Evaluate individual determinants:
det(A^{-1}) = 1 / det(A) = 1 / 5.
det(A^T) = det(A) = 5.
det(adj(A)) = (det(A))^{n - 1} = 5^{3 - 1} = 5^2 = 25.
Step 4: Compute final product:
Result = 8 (1 / 5) 5 * 25
Result = 8 1 25 = 200.
Conclusion: det( 2 A^{-1} A^T * adj(A) ) = 200.
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