Properties of the Transpose of a Matrix

✍️Properties of the transpose of a matrix are mentioned below:

🌞(AT)T = A

🌞(A+B)T = AT + BT

🌞(AB)T = BTAT

Rank of a Matrix

✍️Rank of Matrix is given by the maximum number of linearly independent rows or columns of a matrix. The rank of a matrix is always less than or equal to the total number of rows or columns present in a matrix. A square matrix has linearly independent rows or columns if the matrix is non-singular i.e. determinant is not equal to zero. Since a zero matrix has no linearly independent rows or columns its rank is zero.

✍️Rank of a matrix can be calculated by converting the matrix into Row-Echelon Form. In row echelon form we try to convert all the elements belonging to a row to be zero using Elementary Opeartion on Row. After the operation, the total number of rows which has at least one non-zero element is the rank of the matrix. The rank of the matrix A is represented by ρ(A).

Sets and Relations

🌝What is Relation in Mathematics?

🌝Relations Examples

🌝Representation of Relations

🌝Sets and Relations

🌝Types of Relation

🌝Graphing Relations

🌝What is Relation in Mathematics?

✍️Relation in Mathematics is defined as the relationship between two sets. If we are given two sets set A and set B and set A has a relation with set B then each value of set A is related to a value of set B through some unique relation. Here, set A is called the domain of the relation, and set B is called the range of the relation.

For example if we are given two sets, Set A = {1, 2, 3, 4} and Set B = {1, 4, 9, 16} then the ordered pair {(1, 1), (2, 4), (3, 9), (4, 16)} represents the relation defined as, R, A: → B {(x, y): y = x2: y ϵ B, x ϵ A}. 

🌝Relation Definition:

✍️Relation is defined as the relation between two different sets of information. Suppose we are given two sets containing two different values then a relation defined such that it connects the value of the first set with the value of the second set is called the relation.

Suppose we are given a set A that contains the name of girls of a class and another set B that contains the height of girls then a relation connects set A with set B. In mathematical terms, we can say that,

"A set of ordered pairs is a relation” 

🌝Representation of Relations:

✍️In mathematics or in set theory we can represent the relation using different techniques and the two important ways to represent the set are,

👉Set Builder Notation

👉Roaster Notation

Let’s study them in detail in the article below,

🌝Set Builder Notation:

✍️If are relation between two sets is represented using the logical formula then this type of representation is called the set builder notation.

✍️For example, if we are given two sets set X = {2, 4, 6} and set Y = {4, 8, 12}. Then after observing clearly, we can see that each element of set Y is twice each element of set X the relation between them is,

👉R {(a, b): b is twice of a, a ∈ X, b ∈ Y}

🌝Roaster Form

✍️Roaster form is another way of representing a relation. In roaster form, we use ordered pairs to represent the relation.

✍️For example, if we are given two sets set X = {2, 4, 6} and set Y = {4, 8, 12}. Then the relation between set X and set Y is represented using the relation R such that,

👉R = {(2, 4), (4, 8), (6, 12)}

🌝Types of Sets in Mathematics:

👉Singleton Set

👉Empty Set

👉Finite Set

👉Infinite Set

👉Equal Set

👉Equivalent Set

👉Subset

👉Power Set

👉Universal Set 

👉Disjoint Sets

🌝Sets and Relations:

✍️A well-defined collection of Objects or items or data is known as a set. The objects or data are known as the element. For Example, the boys in a classroom can be put in one set, all integers from 1 to 100 can become one set, and all prime numbers can be called an Infinite set. 

And relations are relationship that connects the values of two sets. So sets and relations are connected to each other. For example, if we are given two sets

👉Set A = {-2, -1, 0, 1, 2}

👉Set B = {2, 3, 4, 5, 6}

Then the relation that connects the two sets, set A and set B,

R = {(-2, 2), (-1, 3), (0, 4), (1, 5), (2, 6)}

🌝Types of Relation:

✍️Various types of relations defined in mathematics are,

    👉Empty Relation

    👉Reflexive Relation

    👉Symmetric Relation

    👉Transitive Relation

    👉Equivalence Relation

    👉Universal Relation

    👉Identity Relation

    👉Inverse Relation

🌝Empty Relation:

✍️A relation R on a set A is called Empty if the set A is an empty set, i.e. any relation where no element of set A is not related to the element of set B then it is called an empty relation. For example, A = {1, 2, 3} and B = {5, 6, 7} where, R = {(x, y) where x + y = 22}, then it is an empty relation.

🌝Reflexive Relation:

✍️A relation R on a set A is called reflexive if (a, a) ∈ R holds for every element a∈ A . i.e. if set A = {a, b} then R = {(a, a), (b, b)} is reflexive relation.

For example, A = {2, 3} then the reflexive relation R on A is,

• R = {(2, 2), (3, 3)}

🌝Symmetric Relation:

✍️A relation R on a set A is called symmetric if (b, a) ∈ R holds when (a, b) ∈ R i.e. The relation R = {(a, b), (b, a)} is a reflexive relation on (a, b)

For example, A = {2, 3} then symmetric relation R on A is,

• R = {(2, 3), (3, 2)}

🌝Transitive Relation:

✍️A relation R on a set A is called transitive if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R for all a,b,c ∈  A i.e.

For example, set A = {1, 2, 3} then transitive relation R on A is,

R = {(1, 2), (2, 3), (1, 3)}

🌝Equivalence Relation:

✍️A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive. i.e. relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)} on set A = {1, 2, 3} is equivalence relation as it is reflexive, symmetric, and transitive.

🌝Universal Relation:

✍️Universal relation is a relation in which all elements of set are mapped to other element of set then it is called universal relation. For example, A = {4, 8, 12} and B = {1, 2, 3} then universal relation is, R = {(x, y) where x > y}

🌝Identity Relation:

✍️Identity relation is a relation defined such all elements in a set are related to itself. It is defined as, I = {(x, x) : for all x ∈ X}.

For example P = {1, 2, 3} then Identity Relation(I) = {(1, 1), (2, 2), (3, 3)}

🌝Inverse Relation:

✍️A relation is called the inverse of any relation if elements of one set is inverse pair of another set. Inverse of a relation R is denoted as R-1. i.e., R-1 = {(y, x): (x, y) ∈ R}.

🌝Graphing Relations:

✍️Relations can be easily represents on the graphs and representing them on graphs is an easy way of explaining them. The ordered pair in a relation represents a coordinate that can be plot on cartesian coordinate system. We can easily graph the relation by following the steps added below,

👉Substitue x with random numerical values in the relation.

👉Find the corresponding y value of the respective x value.

👉Write the ordered pair such that, {(x, y)}

👉Plot these points and join them to find the required curve.

🌝Practice Questions on Relations:

Q1. Check wether the relation R = {(a, b), (b, c), (c, a), (a, a)} is Equivalence relation on set A = {a, b, c}

Q2. Check wether the relation R = {(1, 1), (1, 3), (1, 1), (2, 2)} is Equivalence relation on set A = {1, 2, 3}

Q3. Find the inverse of the relation R = {(1, 1), (2, 4), (3, 9)}

Q4. Find the inverse of the relation R = {(2, 6), (3, 7), (5, 9)}

Determinant of a Matrix with Solved Examples

✍️Determinant of a Matrix is defined as the function that gives the unique output (real number) for every input value of the matrix. Determinant of the matrix is considered the scaling factor that is used for the transformation of a matrix. It is useful for finding the solution of a system of linear equations, the inverse of the square matrix, and others. The determinant of only square matrices exists. 

🌝Content:

👉Definition of Determinant of Matrix

👉Determinant of a 1×1 Matrix

👉Determinant of 2×2 Matrix

👉Determinant of a 3×3 Matrix

👉Determinant of 4×4 Matrix

👉Determinant of Identity Matrix

👉Determinant of Symmetric Matrix

👉Determinant of Skew-Symmetric Matrix

👉Determinant of Inverse Matrix

👉Determinant of Orthogonal Matrix

👉Physical Significance of Determinant

👉Laplace Formula for Determinant

👉Properties of Determinants of Matrix

🌝Definition of Determinant of Matrix:

• Determinant of a Matrix is defined as the sum of products of the elements of any row or column along with their corresponding co-factors. Determinant is defined only for square matrices.

• Determinant of any square matrix of order 2×2, 3×3, 4×4, or n × n, where n is the number of rows or the number of columns. (For a square matrix number of rows and columns are equal). Determinant can also be defined as the function which maps every matrix with the real numbers. 

• For any set S of all square matrices, and R the set of all numbers the function f, f: S → R is defined as f (x) = y, where x ∈ S and y ∈ R, then f (x) is called the determinant of the input matrix.

Let’s take any square matrix A, then the determinant of A is denoted as det A (or) |A|. Determinant is also denoted by the symbol Δ.  

🌝Determinant of Symmetric Matrix:

• A symmetric matrix is a square matrix that is equal to its transpose. In other words, if A is a symmetric matrix, then A = AT. Symmetric matrices have several interesting properties, one of which is that their determinants remain unchanged under transpose. 

Hence, for a symmetric matrix A, we have: det(A)=det(A^T)

• This property simplifies the computation of determinants for symmetric matrices since you can work with either the original matrix or its transpose, whichever is more convenient.

🌝Determinant of Skew-Symmetric Matrix:

• A skew-symmetric (or antisymmetric) matrix is a square matrix whose transpose is equal to its negative. In other words, if A is a skew-symmetric matrix, then A = −AT. Skew-symmetric matrices have interesting properties, one of which is that their determinants have specific values based on the order of the matrix.

• For skew-symmetric matrices of odd order, the determinant is always 0. This is because the determinant of a skew-symmetric matrix is always the square of its eigenvalues, and a non-zero square is always positive. Since the order of the matrix is odd, at least one eigenvalue must be zero, resulting in a determinant of 0.

• For skew-symmetric matrices of even order, the determinant is a non-zero value, which can be calculated based on the elements of the matrix. However, determining the exact value typically involves more complex methods such as cofactor expansion or using properties of determinants.

🌝Determinant of Inverse Matrix:

✍️To understand the determinant of the inverse matrix, let’s first define the inverse of a matrix:

• The inverse of a square matrix A, denoted as A^{-1}, is a matrix such that when it’s multiplied by A, the result is the identity matrix I. Mathematically, if A*A^{−1}=A^{-1}*A = I, then A^{−1} is the inverse of A.

• Now, the determinant of the inverse matrix, denoted as det(A^{−1}), is related to the determinant of the original matrix A. Specifically, it can be expressed by the formula:

det(A^{-1})=1/det(A)

• This formula illustrates an important relationship between the determinants of a matrix and its inverse. If the determinant of A is non-zero, meaning det(A)≠0, then the inverse matrix exists, and its determinant is the reciprocal of the determinant of A. Conversely, if det(A)=0, the matrix A is said to be singular, and it does not have an inverse. 

✍️Here are some key points about the determinant of the inverse matrix:

• Non-Singular Matrices: For non-singular matrices (those with non-zero determinants), their inverses exist, and the determinant of the inverse is the reciprocal of the determinant of the original matrix.

• Singular Matrices: Singular matrices (those with zero determinants) do not have inverses. Attempting to find the inverse of a singular matrix results in an undefined or non-existent inverse.

• Geometric Interpretation: The determinant of a matrix measures how it scales the space. Similarly, the determinant of the inverse matrix measures the scaling effect of the inverse transformation. If the original transformation expands the space, its inverse contraction will be inversely proportional, and vice versa.

🌝Determinant of Orthogonal Matrix

• An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning that the dot product of any two distinct rows or columns equals zero, and the dot product of each row or column with itself equals one. Mathematically, if A is an orthogonal matrix, then AT⋅A=I, where AT denotes the transpose of A and I represents the identity matrix.

• The determinant of an orthogonal matrix has a special property: det(A)=+1 or -1

• The determinant of an orthogonal matrix is either +1+1 or −1−1. This property arises from the fact that the determinant represents the scaling factor of the matrix transformation. Since orthogonal transformations preserve lengths, the determinant must be either positive (for preserving orientation) or negative (for reversing orientation).

• The determinant of an orthogonal matrix being +1,+1 implies that the transformation preserves orientation, while a determinant of −1,−1 indicates a transformation that reverses orientation.

🌝Properties of Determinants of Matrix:

✍️Various Properties of the Determinants of the square matrix are discussed below:

☘️Reflection Property: Value of the determinant remains unchanged even after rows and columns are interchanged. That determinant of a matrix and its transpose remains the same.

☘️Switching Property: If any two rows or columns of a determinant are interchanged, then the sign of the determinant changes.

☘️Scalar Multiplication Property: If each element in a row or column of a matrix A is multiplied by a scalar k, then the determinant of the resulting matrix is k times the determinant of A. Mathematically, if B is the matrix obtained by multiplying each element of a row or column of A by k, then det(B) = k⋅det(A).

☘️Additivity Property: The determinant of the sum of two matrices A and B is equal to the sum of their determinants. Symbolically, det(A+B) = det(A) + det(B). However, this property applies only if the matrices have the same dimensions.

☘️Multiplicative Property: The determinant of the product of two matrices A and B is equal to the product of their determinants. Symbolically, det(AB) = det(A)⋅det(B). However, this property holds true only for square matrices.

☘️Determinant of Transpose: The determinant of a matrix A is equal to the determinant of its transpose AT. Mathematically, det(A) = det(AT).

☘️Repetition Property/Proportionality Property: If any two rows or any two columns of a determinant are identical, then the value of the determinant becomes zero.

☘️Scalar Multiple Property: If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k

☘️Sum Property: If some or all elements of a row or column can be expressed as the sum of two or more terms, then the determinant can also be expressed as the sum of two or more determinants.