The experimental matrix is a table containing all the trials to be performed. It includes the number of trials to be run and the levels assigned to each factor in each trial. The effects matrix is used to calculate the effects. The effects matrix for factorial designs is a Hadamard matrix.

In statistics, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set of objects. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object.

model. matrix creates a design matrix from the description given in terms(object) , using the data in data which must supply variables with the same names as would be created by a call to model. After coercion, all the variables used on the right-hand side of the formula must be logical, integer, numeric or factor.

The hat matrix is a matrix used in regression analysis and analysis of variance. It is defined as the matrix that converts values from the observed variable into estimations obtained with the least squares method.

In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if . For this product to be defined, must necessarily be a square matrix.

Definition 1-13. The rank of a matrix is the maximum number of its linearly independent column vectors (or row vectors). From this definition it is obvious that the rank of a matrix cannot exceed the number of its rows (or columns).

A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank-deficient if it does not have full rank.

The normal form of a matrix A is a matrix N of a pre-assigned special form obtained from A by means of transformations of a prescribed type.

You can see that the determinants of each 3 x 3 sub matrices are equal to zero, which show that the rank of the matrix is not 3. Hence, the rank of the matrix B = 2, which is the order of the largest square sub-matrix with a non zero determinant.

Let us take an indentity matrix or unit matrix of order 3×3. We can see that it is an Echelon Form or triangular Form . Now we know that the number of non zero rows of the reduced echelon form is the rank of the matrix. In our case non zero rows are 3 hence rank of matrix is = 3.

A zero matrix is a matrix in which all of the entries are 0. Some examples are given below. A zero matrix is indicated by O, and a subscript can be added to indicate the dimensions of the matrix if necessary.

The matrix whose every element is zero is called a null or zero matrix and it is denoted by 0. For example, [00] is a zero matrix of order 1 × 2. [00] is a zero or null matrix of order 2 × 1.

Two matrices are called equal matrices if they have the same order or dimension and the corresponding elements are equal. Suppose A and B are the matrices of equal order i × j and aij = bij, then A are B are called equal matrices.

In Mathematics, a scalar matrix is a special kind of diagonal matrix. We can say that the scalar matrix is a diagonal matrix, in which the diagonal contains the same element. A well-known example of the scalar matrix is the identity matrix, in which the diagonal element contains the same value as 1.

A scalar matrix is a special kind of diagonal matrix. It is a diagonal matrix with equal-valued elements along the diagonal. Two examples of a scalar matrix appear below. The identity matrix is also an example of a scalar matrix.

A scalar is an element of a field which is used to define a vector space. Thus, for example, the product of a 1×n matrix and an n×1 matrix, which is formally a 1×1 matrix, is often said to be a scalar. The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.

Scalar, a physical quantity that is completely described by its magnitude; examples of scalars are volume, density, speed, energy, mass, and time. Other quantities, such as force and velocity, have both magnitude and direction and are called vectors.

Scalar multiplication is easy. You just take a regular number (called a “scalar”) and multiply it on every entry in the matrix.

(4) Rectangular Matrix: Rectangular matrix is a type of matrix which has unequal number of rows and columns. Example of rectangular matrix can be given as. , where we have unequal number of rows and columns in a matrix. Number of columns is 2 and number of rows is 3.

What is a Matrix? A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns. The order of the matrix is defined as the number of rows and columns. The entries are the numbers in the matrix and each number is known as an element.

This tutorial is divided into 6 parts to cover the main types of matrices; they are:

• Square Matrix.
• Symmetric Matrix.
• Triangular Matrix.
• Diagonal Matrix.
• Identity Matrix.
• Orthogonal Matrix.

The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

Determinant of rectangular matrix, Radić’s determinant. (2) If a row of A is multiplied by a number k, then the determinant of the resulting matrix is equal to k|A|. (3) Interchanging two rows of A results in changing the sign of the determinant. (4) The determinant |A| can be calculated using the Laplace expansion.

Order of Matrix = Number of Rows x Number of Columns See the below example to understand how to evaluate the order of the matrix. Also, check Determinant of a Matrix. In the above picture, you can see, the matrix has 2 rows and 4 columns. Therefore, the order of the above matrix is 2 x 4.

Multiplication of 2×3 and 3×3 matrices is possible and the result matrix is a 2×3 matrix.

Multiplication of 3×3 and 3×2 matrices is possible and the result matrix is a 3×2 matrix.

Example: This matrix is 2×3 (2 rows by 3 columns): When we do multiplication: The number of columns of the 1st matrix must equal the number of rows of the 2nd matrix. And the result will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.