Dick said:
A linear algebra square matrix is a type of matrix that has an equal number of rows and columns. It is represented by a capital letter and contains numbers or variables. Square matrices are commonly used in linear algebra to represent systems of linear equations and perform various operations such as addition, subtraction, multiplication, and inversion.
A square matrix has several properties that make it unique. These include having an equal number of rows and columns, being symmetric if it is the same when reflected along the main diagonal, being skew-symmetric if its elements change sign when reflected along the main diagonal, and being invertible if it has a unique solution when solving for its inverse.
To multiply two square matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. To find the value of each element in the resulting matrix, you multiply the corresponding row of the first matrix by the corresponding column of the second matrix and then add the products.
The determinant of a square matrix is a scalar value that represents the scaling factor of the matrix. It is calculated by a specific formula and can be used to determine if a matrix is invertible. If the determinant is zero, the matrix is not invertible, and if it is non-zero, the matrix is invertible.
Square matrices have many real-world applications, including computer graphics, engineering, physics, economics, and data analysis. They are used to solve systems of linear equations, perform transformations, and analyze data. For example, in computer graphics, square matrices are used to represent the transformation of 3D objects, and in economics, they are used to model supply and demand systems.