A regular matrix of order n is a matrix that has the same number of rows and columns and its determinant is nonzero (0).
In other words, a regular matrix of order n is a square matrix from which we can obtain the inverse matrix.
Regular array formula
Given a matrix V with the same number of rows (n) and columns (m), that is, m = n, and with a non-zero determinant (0), then we say that V is a regular matrix of order n.
The regular matrix is used as a label for the matrices that meet the conditions to have an inverse matrix.
- The matrix is a square matrix.
The number of rows (n) has to be the same as the number of columns (m). That is, the order of the matrix has to be n given that n = m.
- The matrix has a determinant and this is different from zero (0).
The determinant of the matrix must be nonzero (0) because it is used as the denominator in the inverse matrix formula.
Is matrix D a square and invertible matrix?
- We check if matrix D meets the requirements to be a regular matrix.
- Is matrix D a square matrix?
The number of columns in matrix D is different from the number of rows since there are 2 rows and 3 columns. Therefore, matrix D is not a square matrix, nor is it a regular matrix.
The first condition to be a regular matrix (square matrix condition) is a necessary and sufficient requirement since if it is not fulfilled it directly implies that the matrix is not a regular matrix and therefore we will not be able to calculate its determinant.
- Is matrix D invertible?
Since matrix D is not square, we cannot calculate its determinant and decide if it is different from or equal to zero (0).
Regular matrix of order 2
Is matrix U a square and invertible matrix?
- We check if matrix U meets the requirements to be a regular matrix.
- Is matrix U a square matrix?
The number of rows and the number of columns match in matrix U. So the matrix U is a square matrix of order 2.
- Is the matrix U invertible?
First we will have to calculate the determinant of the matrix and then check that it is different from zero (0).
- Determinant of matrix U :
- Check that the matrix U is invertible:
So the matrix U is a regular matrix since it is a square and invertible matrix.
Inverse matrix of order 2