Introduction to Matrices: Types, Operations, and Properties

Gauss

Calculating the Inverse of a Matrix

Let A = (a_ij) be a square matrix of order n. To calculate the inverse of A, denoted as A^-1, follow these steps:

Step 1: Building the Augmented Matrix

Build the n x 2n matrix M = (A | I), where A is in the left half of M and the identity matrix I is on the right.

Step 2: Gaussian Elimination

Keep the first row of M as it is. Below the first main diagonal element, a₁₁ (which we’ll call the pivot), put zeros. Then operate as shown in the following example.

Example: Finding the Inverse of a 3×3 Matrix

Step 1: Building the Augmented Matrix

Consider an arbitrary 3 x 3 matrix.

Step 2: Gaussian Elimination

The next step is the same as above, but this time you take as a pivot the second term on the diagonal.

For the last term on the diagonal, proceed as before, putting zeros above the new pivot. Note that as you use each pivot, the matrix A on the left half of M becomes a triangular matrix.

Once you’ve completed all the steps, the left half of the matrix M becomes a diagonal matrix. At this point, you must transform the left half into the identity matrix, if necessary by dividing the rows of M by a scalar.

Example: Finding the Inverse of a Specific Matrix

Suppose we want to find the inverse of a specific matrix.

First, we build the matrix M = (A | I).

The left half of M is triangular, hence A is invertible. If there had been a row with all zeros in the left half (A) of M, the operation would be over (A is not invertible).

Then we take as a pivot a₃₃, put zeros above it, and continue to operate until we have a diagonal matrix on the left.

Since the matrix on the left half is diagonal, there is no more to operate. We transform the diagonal matrix into an identity matrix; this requires dividing the second row by -1.

The matrix on the right half of M is precisely the inverse of A.

Checking the Result

To check if the result is correct, we multiply AA^-1, which should result in the identity matrix I.

Check:

AA^-1 = I

Types of Matrices

Matrix Row

A row matrix is formed by a single row.

Matrix Column

A column matrix has a single column.

Rectangular Matrix

A rectangular matrix has a different number of rows and columns, and its dimension is m x n.

Square Matrix

A square matrix has the same number of rows and columns.

The elements of the form a_ii are the main diagonal.

The secondary diagonal consists of elements with the form i + j = n + 1.

Zero Matrix

In a zero matrix, all elements are zeros.

Upper Triangular Matrix

In an upper triangular matrix, elements located below the main diagonal are zeros.

Lower Triangular Matrix

In a lower triangular matrix, elements above the main diagonal are zeros.

Diagonal Matrix

In a diagonal matrix, all elements above and below the main diagonal are zero.

Scalar Matrix

A scalar matrix is a diagonal matrix in which the main diagonal elements are equal.

Identity Matrix or Unit Matrix

An identity matrix is a diagonal matrix in which the main diagonal elements are equal to 1.

Matrix Transpose

Given a matrix A, the matrix transpose of A, denoted by A^t, is obtained by interchanging rows and columns.

(A^t)^t = A

(A + B)^t = A^t + B^t

(λ · A)^t = λ · A^t

(AB)^t = B^t · A^t

Regular Matrix

A regular matrix is a square matrix that has an inverse.

Singular Matrix

A singular matrix has no inverse matrix.

Idempotent Matrix

A matrix A is idempotent if:

A² = A

Involutional Matrix

A matrix A is involutive if:

A² = I

Symmetric Matrix

A symmetric matrix is a square matrix that verifies:

A = A^t

Antisymmetric or Skew-Symmetric Matrix

An antisymmetric or skew-symmetric matrix is a square matrix that verifies:

A = –A^t

Orthogonal Matrix

A matrix is orthogonal if it satisfies:

A · A^t = I

Matrix Addition

Given two matrices of the same dimension, A = (a_ij) and B = (b_ij), the matrix sum is defined as: A + B = (a_ij + b_ij). That is, the elements of the resulting matrix are obtained by summing the elements of the two matrices that occupy the same position.

Matrix addition is a binary operation with the following properties:
PROPERTIES:

· Associative: A + (B + C) = (A + B) + C
· Commutative: A + B = B + A
· Neutral Element (zero matrix 0_{m x n}): 0 + A = A + 0 = A
· Additive Inverse (opposite matrix –A): A + (-A) = (-A) + A = 0

Matrix Multiplication

Two matrices A and B can be multiplied if the number of columns of A matches the number of rows of B.

M_{m x n} x R_{n x p} = M_{m x p}

The element c_ij of the product matrix is obtained by multiplying each element in row i of matrix A by each element of column j of matrix B and adding them.

Product Properties of Matrices

Associative Property

A · (B x C) = (AB)C

Neutral Element

A · I = A

Where I is the identity matrix of the same order as the matrix A.

Non-Commutative Property

A · B ≠ B · A

Distributive Property

A · (B + C) = A · B + A · C