Linear Algebra: Matrices, Determinants, and Gaussian Methods

Transposed and Symmetric Matrices

When we swap rows and columns, we obtain a transposed matrix.

A symmetric square matrix is a specific case where the matrix equals its transpose. For example: F (121/224/146) where F^t = (121/224/146).

Systems of Linear Equations

A way to represent a system of linear equations with unknowns is a System:

• ax + by = c
• dx + ey = f

Opposite Matrices and Sum Properties

The opposite of a given matrix is the result of replacing each element by its opposite. The opposite of A is -A.

The properties of the matrix sum include:

• Associative Property
• Commutative Property
• Null Element
• Matrix Opposed

Properties of Determinants

The properties of determinants and their examples are as follows:

• The determinant of a square matrix equals the determinant of its transposed matrix.
• If the elements of each row or column of a matrix are multiplied by a number, the determinant of the matrix is multiplied by that number.
• If all the lines of a matrix of order n are multiplied by the same number t, the determinant of the matrix is multiplied by tⁿ.
• The determinant of the product of two square matrices is the product of the determinants of both matrices.
• If two lines of a square matrix are interchanged, its determinant changes sign.
• If a line of a square matrix is a linear combination of the remaining lines (i.e., it is the result of adding elements from other lines multiplied by real numbers), its determinant is zero.
• Therefore, a direct consequence of this property is that if a matrix has a line of zeros, its determinant is zero.
• If a linear combination of the remaining lines is added to the elements of each row of a square matrix, the determinant does not vary.

Gauss and Gauss-Jordan Methods

It is recommended to use the strategy of pivoting in any computer implementation of these methods. With the help of this strategy, one can reduce rounding errors and avoid problems such as division by zero.

Although in all other respects they are equal, Gaussian elimination is preferable to Gauss-Jordan, since the former is 50% faster. However, the Gauss-Jordan method is still useful because it can be modified slightly to obtain the inverse matrix as a bonus during calculations. Although these elimination methods have great utility, the use of the entire coefficient matrix can be a very important factor when dealing with very large and dispersed systems.