Matricies & Linear Algebra

Transpose of Matrix A

Swap the rows and columns (Row 1 becomes Col 1)

Matrix Dimension Order

Rows x Columns (e.g. 2 rows 3 cols is 2x3)

Matrix Addition Commutativity (A+B=B+A)

True

Matrix Multiplication Commutativity (AB=BA)

False

Scalar Multiplication (cA=Ac)

True

Valid Elementary Row Operations

1. Swap Rows 2. Scale Row (multiply by non-zero) 3. Add multiple of one row to another

Condition for "Infinite Solutions" in a system

A row of all zeros (0=0) with fewer pivots than variables

Condition for "No Solution" in a system

A row where zero equals a non-zero number (e.g. 0=5)

Rank of a Matrix

The number of non-zero rows after row reduction

Cramer's Rule Formula for xi​

Di​/D (Determinant of modified matrix / Determinant of coefficient matrix)

Condition for Matrix Inverse to exist

The Determinant must be non-zero

Eigenvalue Definition

The scalar λ in the equation Ax=λx

Eigenvector Definition

The non-zero vector x that does not change direction when multiplied by A

Characteristic Equation for eigenvalues

det(A−λI)=0

Eigenvalues of a Triangular Matrix

The entries on the main diagonal

Matrix Condition Number κ(A)≈1

Well-Conditioned

Matrix Condition Number κ(A) is very large

Ill-Conditioned

Diagonal Dominance

When the absolute value of the diagonal element is larger than the sum of all other elements in that row

If a matrix is NOT diagonally dominant…

Convergence of iterative methods (Jacobi/Gauss-Seidel) is not guaranteed

Frobenius Norm

Square root of the sum of squares of all elements

Row Sum Norm

The maximum absolute row sum

Column Sum Norm

The maximum absolute column sum

Multiplication Validity (A is 3×2

B is 3×2)

Multiplication Validity (A is 3×3

B is 3×2)