This advanced section is available to members.
Upgrade membership7.3Source-backedMemberEstimated reading time: 10 min
7.3 Transpose, column operations, and Cramer's rule
Use transpose and column operations to read determinants from a second angle, then finish the chapter with adjoints, inverse formulas, and Cramer's rule.
Note collections
MATH1030: Linear algebra I
Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.
Chapter 1
Systems of equations
Learn to read equations as full solution sets.
Chapter 2
Matrices and elimination
Build matrix intuition and use row reduction with purpose.
Chapter 3
Matrix algebra
Matrix multiplication, transpose, and structural matrix notation.
Chapter 4
Solution structure
Homogeneous systems, null spaces, and the shape of full solution sets.
Chapter 5
Invertibility
Understand when a matrix can be undone and why that matters.
Chapter 6
Vector spaces
Move from matrix procedures to the structure of spaces, span, independence, and basis.
Chapter 7
Determinants
Determinants, cofactor formulas, and the structural algebra that connects row operations, transpose, and invertibility.
Chapter 8
Eigenvalues and diagonalization
Eigenvalues, eigenspaces, similarity, and diagonalization as the next structural layer after determinants.
Chapter 9
Inner products and orthogonality
Inner products, orthogonality, orthonormal bases, and Gram-Schmidt as the geometric layer after eigenvalues.