Matrix is a rectangular array of numbers, symbols, points, or characters each belonging to a specific row and column. A matrix is identified by its order which is given in the form of rows ⨯ and columns. The numbers, symbols, points, or characters present inside a matrix are called the elements of a matrix. The location of each element is given by the row and column it belongs to.
Matrices are rectangular arrays of numbers, symbols, or characters where all of these elements are arranged in each row and column. An array is a collection of items arranged at different locations.
Let’s assume points are arranged in space each belonging to a specific location then an array of points is formed. This array of points is called a matrix. The items contained in a matrix are called Elements of the Matrix. Each matrix has a finite number of rows and columns and each element belongs to these rows and columns only. The number of rows and columns present in a matrix determines the order of the matrix. Let’s say a matrix has 3 rows and 2 columns then the order of the matrix is given as 3⨯2.
A rectangular array of numbers, symbols, or characters is called a Matrix. Matrices are identified by their order. The order of the matrices is given in the form of a number of rows ⨯ number of columns. A matrix is represented as [P]m⨯n where P is the matrix, m is the number of rows and n is the number of columns. Matrices in maths are useful in solving numerous problems of linear equations and many more.
Order of a Matrix tells about the number of rows and columns present in a matrix. Order of a matrix is represented as the number of rows times the number of columns. Let’s say if a matrix has 4 rows and 5 columns then the order of the matrix will be 4⨯5. Always remember that the first number in the order signifies the number of rows present in the matrix and the second number signifies the number of columns in the matrix.
Examples of matrices are mentioned below:
Example: [1234]2×2, [1−123264−25]3×3
Matrices undergo various mathematical operations such as addition, subtraction, scalar multiplication, and multiplication. These operations are performed between the elements of two matrices to give an equivalent matrix that contains the elements which are obtained as a result of the operation between elements of two matrices. Let’s learn the operation of matrices.
In addition of matrices, the elements of two matrices are added to yield a matrix that contains elements obtained as the sum of two matrices. The addition of matrices is performed between two matrices of the same order.
Example: Find the sum of [1245]and [2367]
Solution:
Here, we have A = [1245]and B = [2367]
A + B = [1245]+ [2367]
⇒ A + B = [1+22+34+65+7]= [351012]
Subtraction of Matrices is the difference between the elements of two matrices of the same order to give an equivalent matrix of the same order whose elements are equal to the difference of elements of two matrices. The subtraction of two matrices can be represented in terms of the addition of two matrices. Let’s say we have to subtract matrix B from matrix A then we can write A – B. We can also rewrite it as A + (-B). Let’s solve an example
Example: Subtract [1245]from [2367].
Let us assume A = [2367]and B = [1245]
A – B = [2367]– [1245]
⇒ A – B = [2–13–26–47–5]= [1122]
Scalar Multiplication of matrices refers to the multiplication of each term of a matrix with a scalar term. If a scalar let’s ‘k’ is multiplied by a matrix then the equivalent matrix will contain elements equal to the product of the scalar and the element of the original matrix. Let’s see an example:
Example: Multiply 3 with [1245].
3[A] = [3×13×23×43×5]
⇒ 3[A] = [361215]
In the multiplication of matrices, two matrices are multiplied to yield a single equivalent matrix. The multiplication is performed in the manner that the elements of the row of the first matrix multiply with the elements of the columns of the second matrix and the product of elements are added to yield a single element of the equivalent matrix. If a matrix [A]i⨯j is multiplied with matrix [B]j⨯k then the product is given as [AB]i⨯k.
Let’s see an example.
Example: Find the product of [1245]and [2367]
Solution:
Let A = [1245]and B = [2367]
⇒ AB = [1245][2367]
⇒ AB = [1×2+2×61×3+2×74×2+5×64×3+5×7]
⇒ AB=[14173847]
Properties followed by Multiplication and Addition of Matrices is listed below:
Transpose of Matrix is basically the rearrangement of row elements in column and column elements in a row to yield an equivalent matrix. A matrix in which the elements of the row of the original matrix are arranged in columns or vice versa is called Transpose Matrix. The transpose matrix is represented as AT. if A = [aij]mxn , then AT = [bij]nxm where bij = aji.
Let’s see an example:
Example: Find the transpose of [18173847] .
Solution:
Let A = [18173847]
⇒ AT = [18381747]
Properties of the Transpose of a Matrix
Properties of the transpose of a matrix are mentioned below:
Trace of a Matrix is the sum of the principal diagonal elements of a square matrix. Trace of a matrix is only found in the case of a square matrix because diagonal elements exist only in square matrices. Let’s see an example.
Example: Find the trace of the matrix [123456789]
Solution:
Let us assume A = [123456789]
Trace(A) = 1 + 5 + 9 = 15
Based on the number of rows and columns present and the special characteristics shown, matrices are classified into various types.
Note: Every Square Matrix can uniquely be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. A = 1/2 (AT + A) + 1/2 (A – AT).
Learn More, Types of Matrices
Determinant of a matrix is a number associated with that square matrix. The determinant of a matrix can only be calculated for a square matrix. It is represented by |A|. The determinant of a matrix is calculated by adding the product of the elements of a matrix with their cofactors.
Determinant of a Matrix
Let’s see how to find the determinant of a square matrix.
Example 1: How to find the determinant of a 2⨯2 square matrix?
Let say we have matrix A = [abcd]
Then, determinant is of A is |A| = ad – bc
Example 2: How to find the determinant of a 3⨯3 square matrix?
Let’s say we have a 3⨯3 matrix A = [abcdefghi]
Then |A| = a(-1)1+1∣efhi∣+ b(-1)1+2∣dfgi∣ + c(-1)1+3∣degh∣
Minor of a matrix for an element is given by the determinant of a matrix obtained after deleting the row and column to which the particular element belongs to. Minor of Matrix is represented by Mij. Let’s see an example.
Example: Find the minor of the matrix [abcdefghi]for the element ‘a’.
Minor of element ‘a’ is given as M12 = ∣efhi∣
Cofactor of a matrix is found by multiplying the minor of the matrix for a given element by (-1)i+j. Cofactor of a Matrix is represented as Cij. Hence, the relation between the minor and cofactor of a matrix is given as Mij = (-1)i+jMij. If we arrange all the cofactor obtained for an element then we get a cofactor matrix given as C = [c11c12c13c21c22c23c31c32c33]
Learn More, Minors and Cofactors
Adjoint is calculated for a square matrix. Adjoint of a matrix is the transpose of the cofactor of the matrix. The Adjoint of a Matrix is thus expressed as adj(A) = CT where C is the Cofactor Matrix.
Let’s say for example we have matrix
A=[a1b1c1a2b2c2a3b3c3]
then
adj(A)= [A1B1C1A2B2C2A3B3C3]T⇒adj(A)=[A1A2A3B1B2B3C1C2C3]
where,
[A1B1C1A2B2C2A3B3C3]is cofactor of Matrix A.
Properties of the Adjoint of a matrix are mentioned below:
Where, “n = number of rows = number of columns”
A matrix is said to be an inverse of matrix ‘A’ if the matrix is raised to power -1 i.e. A-1. The inverse is only calculated for a square matrix whose determinant is non-zero. The formula for the inverse of a matrix is given as:
A-1 = adj(A)/det(A) = (1/|A|)(Adj A), where |A| should not be equal to zero, which means matrix A should be non-singular.
Elementary Operations on Matrices are performed to solve the linear equation and to find the inverse of a matrix. Elementary operations are between rows and between columns. There are three types of elementary operations performed for rows and columns. These operations are mentioned below:
Elementary operations on rows include:
Elementary operations on columns include:
A matrix formed by combining columns of two matrices is called Augmented Matrix. An augmented matrix is used to perform elementary row operations, solve a linear equation, and find the inverse of a matrix. Let us understand through an example.
Let’s say we have a matrix A = [a1b1c1a2b2c2a3b3c3], X = [xyz]and B = [p1p2p3]then augmented matrix is formed between A and B. The augmented matrix for A and B is given as
[A|B] = [a1b1c1p1a2b2c2p2a3b3c3p3]
Matrices are used to solve linear equations. To solve linear equations we need to make three matrices. The first matrix is of coefficients, the second matrix is of variables and the third matrix is of constants. Let’s understand it through an example.
Let’s say we have two equations given as a1x + b1y = c1 and a2x + b2y = c2. In this case, we will form the first matrix of coefficient let’s say A = [a1b1a2b2], the second matrix is of variables let’s say X = [xy]and the third matrix is of coefficient B = [c1c2]then the matrix equation is given as
AX = B
⇒ X = A-1B
where,
Hence we can see that the value of variable X can be calculated by multiplying the inverse of matrix A with B and then equalizing the equivalent product of two matrices with matrix X.
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).
Eigen Values are the set of scalar associated with the linear equation in matrix form. Eigenvalues are also called characteristic roots of the matrices. The vectors that are formed by using the eigenvalue to tell the direction at that points are called Eigenvectors. Eigenvalues change the magnitude of eigenvectors. Like any vector, Eigenvector doesn’t change with linear transformation.
For a Square Matrix A of order ‘n’ another square matrix A – λI is formed of the same order, where I is the Identity Matrix and λ is the eigenvalue. The eigenvalue λ satisfies an equation Av = λv where v is a non-zero vector.
Learn more about Eigenvalues and Eigenvectors at our website.
The basic formula for the matrices has been discussed below: