Linear Algebra - Lecture 01 Basics of Determinants

Vocabulary flashcards covering basic concepts of matrices and determinants based on Lecture 01 by Dr. Puneet Sharma Sir.

Matrix

A rectangular arrangement of $m \times n$ numbers, represented as $A_{m \times n} = [a_{ij}]$ where $1 \le i \le m$ and $1 \le j \le n$.

Row Matrix

A matrix consisting of a single row, having an order of $1 \times n$, written as $[a_{11}, a_{12}, a_{13}, \dots, a_{1n}]$.

Column Matrix

A matrix consisting of a single column, having an order of $m \times 1$.

Square Matrix

A matrix in which the number of rows is equal to the number of columns.

Trace of a Matrix

The sum of the diagonal elements of a square matrix, denoted as $\text{Trace}(A) = \sum a_{ii}$.

Singular Matrix

A matrix $A$ whose determinant is equal to zero, i.e., $|A| = 0$.

Elementary Row Operations

Transformations applied to matrix rows including $R_i \leftrightarrow R_j$, $R_i \rightarrow k R_i$, and $R_i \rightarrow R_i + k R_j$, where $k$ is any scalar or constant.

Block Matrix

A matrix partitioned into sub-matrices where, if non-zero blocks exist only at diagonal positions, the determinant is the product of the determinants of the diagonal sub-matrices ($|A| = |A_1| \cdot |A_2| \cdot |A_3|$).