Matrices Notes PDF

In Matrices 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...

In Matrices 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 important for students of class 12 and are also of great importance in engineering mathematics. In this introductory article on matrices, we will learn about the types of matrices, the transpose of matrices, the rank of matrices, the adjoint and inverse of matrices, the determinants of matrices, and many more in detail. Table of Content What are Matrices? Matrices Definition Order of Matrix Matrices Examples Operation on Matrices Addition of Matrices Scalar Multiplication of Matrices Multiplication of Matrices Properties of Matrix Addition and Multiplication Transpose of Matrix Trace of Matrix Types of Matrices Determinant of a Matrix Inverse of a Matrix Solving Linear Equation Using Matrices Rank of a Matrix Eigen Value and Eigen Vectors of Matrices Matrices Formulas Unsolved Problems What are Matrices? 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. Matrices Definition 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 Matrix 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. Matrices Examples Example: 2×22×2, [1−123264−25]3×3134−12−22653×3 ​ ​ ​ ​ ​ ​ ​ ​ Operation on Matrices 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. Addition 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 and ​ ​ ​ Solution: Here, we have A = and B = ​ ​ ​ A + B = + ​ ​ ​ ​ ⇒ A + B = [1+22+34+65+7][1+24+62+35+7]= ​ ​ ​ ​ Subtraction of Matrices 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 from. ​ ​ ​ ​ Let us assume A = and B = ​ ​ ​ ​ A – B = – ​ ​ ​ ​ ⇒ A – B = [2–13–26–47–5][2–16–43–27–5]= ​ ​ ​ ​ Scalar Multiplication of Matrices 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. ​ ​ 3[A] = [3×13×23×43×5][3×13×43×23×5] ​ ​ ⇒ 3[A] = ​ ​ Multiplication of Matrices 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 and ​ ​ ​ ​ Solution: Let A = and B = ​ ​ ​ ​ ⇒ AB = ​ ​ ​ ​ ⇒ AB = [1×2+2×61×3+2×74×2+5×64×3+5×7][1×2+2×64×2+5×61×3+2×74×3+5×7] ​ ​ ⇒ AB=AB= ​ ​ Properties of Matrix Addition and Multiplication Properties followed by Multiplication and Addition of Matrices is listed below: A + B = B + A (Commutative) (A + B) + C = A + (B + C) (Associative) AB ≠ BA (Not Commutative) (AB) C = A (BC) (Associative) A (B+C) = AB + AC (Distributive) Transpose of Matrix 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. ​ ​ Solution: Let A = ​ ​ ⇒ AT = ​ ​ 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 Trace of Matrix 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 147258369 ​ ​ ​ ​ Solution: Let us assume A = 147258369 ​ ​ ​ ​ Trace(A) = 1 + 5 + 9 = 15 Types of Matrices Based on the number of rows and columns present and the special characteristics shown, matrices are classified into various types. Row Matrix: A Matrix in which there is only one row and no column is called Row Matrix. Column Matrix: A Matrix in which there is only one column and now row is called a Column Matrix. Horizontal Matrix: A Matrix in which the number of rows is less than the number of columns is called a Horizontal Matrix. Vertical Matrix: A Matrix in which the number of columns is less than the number of rows is called a Vertical Matrix. Rectangular Matrix: A Matrix in which the number of rows and columns are unequal is called a Rectangular Matrix. Square Matrix: A matrix in which the number of rows and columns are the same is called a Square Matrix.  Diagonal Matrix: A square matrix in which the non-diagonal elements are zero is called a Diagonal Matrix. Zero or Null Matrix: A matrix whose all elements are zero is called a Zero Matrix. A zero matrix is also called as Null Matrix. Unit or Identity Matrix: A diagonal matrix whose all diagonal elements are 1 is called a Unit Matrix. A unit matrix is also called an Identity matrix. An identity matrix is represented by I. Symmetric matrix: A square matrix is said to be symmetric if the transpose of the original matrix is equal to its original matrix. i.e. (AT) = A. Skew-symmetric Matrix: A skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative i.e. (AT) = -A. Orthogonal Matrix: A matrix is said to be orthogonal if AAT = ATA = I Idempotent Matrix:A matrix is said to be idempotent if A2 = A Involutory Matrix:A matrix is said to be Involutory if A2 = I. Upper Triangular Matrix: A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix Lower Triangular Matrix: A square matrix in which all the elements above the diagonal are zero is known as the lower triangular matrix Singular Matrix: A square matrix is said to be a singular matrix if its determinant is zero i.e. |A|=0 Nonsingular Matrix: A square matrix is said to be a non-singular matrix if its determinant is non-zero. 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). Determinant of a Matrix 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][acbd] ​ ​ 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]adgbehcfi ​ ​ ​ ​ Then |A| = a(-1)1+1∣efhi∣ehfi+ b(-1)1+2∣dfgi∣dgfi + c(-1)1+3∣degh∣dgeh ​ ​ ​ ​ ​ ​ ​ ​ ​ Minor of a Matrix 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]adgbehcfifor the element ‘a’. ​ ​ ​ Minor of element ‘a’ is given as M11 = ∣efhi∣ehfi ​ ​ ​ Cofactor of Matrix 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]c11c21c31c12c22c32c13c23c33 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Adjoint of a Matrix 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]A=a1a2a3b1b2b3c1c2c3 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ then adj(A)= [A1B1C1A2B2C2A3B3C3]T⇒adj(A)=[A1A2A3B1B2B3C1C2C3]adj(A)= A1A2A3B1B2B3C1C2 ​ ​ ​ ​ ​ ​ ​ ​ C3T⇒adj(A)=A1B1C1A2B2C2A3B3C3 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ where, [A1B1C1A2B2C2A3B3C3]A1A2A3B1B2B3C1C2C3is cofactor of Matrix A. ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Properties of Adjoint of Matrix Properties of the Adjoint of a matrix are mentioned below: A(Adj A) = (Adj A) A = |A| In Adj(AB) = (Adj B). (Adj A) |Adj A| = |A|n-1 Adj(kA) = kn-1 Adj(A) |adj(adj(A))| = ∣A∣(n−1)2∣A∣(n−1)2 adj(adj(A)) = |A|(n-2) × A If A = [L,M,N] then adj(A) = [MN, LN, LM] adj(I) = I {where I is Identity Matrix} Where, “n = number of rows = number of columns” Inverse of a Matrix 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. Properties Inverse of Matrix (A-1)-1 = A (AB)-1 = B-1A-1 only a non-singular square matrix can have an inverse. Elementary Operation on Matrices 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: Interchanging two rows Multiplying a row by a non-zero number Adding two rows Elementary operations on columns include: Interchanging two columns Multiplying a column by a non-zero number Adding two columns Augmented Matrix 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]a1a2a3b1b2b3c1c2c3, X = [xyz]xyzand B ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ = [p1p2p3]p1p2p3then augmented matrix is formed between A and B. The augmented ​ ​ ​ ​ matrix for A and B is given as [A|B] = [a1b1c1p1a2b2c2p2a3b3c3p3]a1a2a3b1b2b3c1c2c3p1p2p3 ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Solving Linear Equation Using Matrices 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][a1a2b1b2], the ​ ​ ​ second matrix is of variables let’s say X = [xy][xy]and the third matrix is of coefficient ​ B = [c1c2][c1c2]then the matrix equation is given as ​ ​ AX = B ⇒ X = A-1B where, A is Coefficient Matrix X is Variable Matrix B is Constant Matrix 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.