A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Think of it like a table or grid where each position holds a value. For example, this three by two matrix has three rows going horizontally and two columns going vertically.
Matrix dimensions tell us the size of a matrix and are written as rows times columns. A two by three matrix has two rows and three columns. A four by one matrix has four rows and one column, making it a column vector. A one by five matrix has one row and five columns, making it a row vector.
Each number in a matrix is called an element or entry. We use subscript notation to identify specific elements. The subscript a i j refers to the element in row i and column j. For example, a subscript 2 3 represents the element in the second row and third column of the matrix.
There are several special types of matrices. A square matrix has the same number of rows and columns. A zero matrix has all elements equal to zero. An identity matrix is a special square matrix with ones on the main diagonal and zeros everywhere else. These special matrices have important properties in linear algebra.
To summarize what we have learned about matrices: A matrix is a rectangular array of numbers arranged in rows and columns. We describe matrix size using dimensions written as rows times columns. Each element has a specific position identified by its row and column. There are special types like square matrices, zero matrices, and identity matrices. Matrices are fundamental mathematical tools used extensively in science, engineering, and data analysis.