Math 410 Study Guide: Matrix Operations, Inverses, LU Factorization, and Determinants

Vocabulary practice flashcards covering fundamental definitions, matrix properties, inverses, LU decomposition, and determinants based on Sections 2.1, 2.2, 2.3, 2.5, and 3.1.

Columns of $AB$

Linear combinations of the columns of $A$, using the entries from the corresponding column of $B$ as weights (represented column-wise as $AB = \begin{pmatrix} Ab_1 & Ab_2 & \dots & Ab_p \end{pmatrix}$).

Rows of $AB$

The rows of the product matrix, each formed by taking the corresponding row of $A$ and multiplying it on the right by $B$.

Transpose of a Product

The matrix identity stating that the transpose of a product equals the product of the transposes in reverse order: $(AB)^T = B^T A^T$.

Associativity of Matrix Multiplication

The matrix multiplication property $(AB)C = A(BC)$, showing that products can be regrouped, though matrix multiplication remains non-commutative.

Invertible Matrix ($A^{-1}$)

A square matrix $A$ for which there exists a matrix $B$ satisfying both equations $AB = I$ and $BA = I$, denoted $B = A^{-1}$.

Inverse of a Product

The algebraic property stating that the product of invertible matrices is invertible, with its inverse reversing the order of factors: $(AB)^{-1} = B^{-1} A^{-1}$.

$2 \times 2$ Invertibility Condition

A $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is invertible if and only if $ad - bc \neq 0$. If $ad = bc$, the determinant denominator is zero and $A$ is not invertible.

Elementary Matrix

A square matrix that is always invertible because each corresponding elementary row operation possesses a reverse row operation.

Invertible Matrix Column Spanning

For an $n \times n$ invertible matrix $A$, the columns span $\mathbb{R}^n$ because the system $Ax = b$ is consistent for every $b \in \mathbb{R}^n$ with the solution given by $x = A^{-1}b$.

Trivial Solution Pivot Implication

The condition where $Ax = 0$ has only the trivial solution, which guarantees that there are no free variables and therefore $A$ has a pivot in every column.

Transpose Invertibility Equivalence

The equivalence establishing that a square matrix $A$ is invertible if and only if its transpose $A^T$ is invertible.

Matrix $U$ (in $A = LU$ Factorization)

The upper triangular echelon form matrix obtained from $A$ by row elimination.

Matrix $L$ (in $A = LU$ Factorization)

The unit lower triangular matrix that stores the elimination multipliers used during the row reduction of $A$ to $U$.

Basic LU Factorization Assumption

The foundational requirement that the matrix $A$ can be reduced to echelon form using elimination without requiring any row interchanges.

$(i,j)$-Cofactor ($C_{ij}$)

The signed scalar value defined by $C_{ij} = (-1)^{i+j} M_{ij}$, where $M_{ij}$ is the minor obtained by taking the determinant of the submatrix resulting from deleting row $i$ and column $j$ of $A$.

Determinant of a Triangular Matrix

The determinant of an upper or lower triangular matrix, calculated explicitly as the product of the entries on its main diagonal.