Matrix and Determinant - Inverse of a matrix

  • A matrix can have an inverse if it is a square matrix (i.e., the number of rows is equal to the number of columns).
  • The inverse of a matrix A is denoted by A^-1.
  • For a matrix to have an inverse, its determinant must be non-zero. Example: Consider the matrix A = [3 2; 1 4] Determinant of A = 34 - 21 = 10 Since the determinant is non-zero, the matrix A has an inverse. Equation: To find the inverse of a matrix A, we can use the equation: A^-1 = (1/det(A)) * adj(A) Where adj(A) represents the adjugate of matrix A. Example: Let’s find the inverse of matrix A by using the above equation: A = [3 2; 1 4] det(A) = 10 adj(A) = [4 -2; -1 3] A^-1 = (1/10) * [4 -2; -1 3] = [2/5 -1/5; -1/10 3/10] Note: The product of a matrix and its inverse is the identity matrix.

Matrix and Determinant - Inverse of a matrix (continued)

Properties of Inverse:

  1. If A has an inverse A^-1, then (A^-1)^-1 = A.
  1. If A and B are invertible matrices, then AB is also invertible and (AB)^-1 = B^-1A^-1.
  1. If A is an invertible matrix, then (A^T)^-1 = (A^-1)^T. Example: Consider matrices A = [2 3; 5 7] and B = [1 2; 3 5]. Let’s find the inverse of A and B: A = [2 3; 5 7] det(A) = 27 - 35 = -1 (non-zero) A^-1 = (1/-1) * [7 -3; -5 2] = [-7 3; 5 -2] B = [1 2; 3 5] det(B) = 15 - 23 = -1 (non-zero) B^-1 = (1/-1) * [5 -2; -3 1] = [-5 2; 3 -1] Now, let’s find the inverse of AB: AB = [2 3; 5 7] * [1 2; 3 5] = [11 20; 26 47] det(AB) = 1147 - 2026 = 1 (non-zero) (AB)^-1 = (1/1) * [47 -20; -26 11] = [47 -20; -26 11] Therefore, (AB)^-1 = B^-1A^-1 = [-5 2; 3 -1] * [-7 3; 5 -2] = [11 20; 26 47] = AB.

Matrix and Determinant - Inverse of a matrix (continued)

Method to find the inverse of a matrix:

  1. Write the given matrix with its elements.
  1. Find the determinant of the given matrix.
  1. If the determinant is non-zero, proceed further.
  1. Find the adjugate of the given matrix.
  1. Multiply the adjugate matrix with the reciprocal of the determinant. Example: Let’s find the inverse of matrix P = [4 7; 2 3]: P = [4 7; 2 3] det(P) = 43 - 72 = 2 (non-zero) adj(P) = [3 -7; -2 4] P^-1 = (1/2) * [3 -7; -2 4] = [3/2 -7/2; -1 2] Therefore, the inverse of matrix P is P^-1 = [3/2 -7/2; -1 2].

Matrix and Determinant - Inverse of a matrix (continued)

Conditions for Inversibility:

  1. A matrix has only one inverse.
  1. If a matrix doesn’t have an inverse, it is called a singular matrix.
  1. If a matrix has an inverse, it is called a non-singular matrix or invertible matrix.
  1. Non-square matrices do not have inverses. Example: Consider the matrix Q = [1 2 3; 4 5 6]. Since Q is not a square matrix, it does not have an inverse.

Matrix and Determinant - Identity Matrix and Inverse

  • The identity matrix, denoted by I, is a square matrix in which all the diagonal elements are 1 and all other elements are 0.
  • The identity matrix plays a similar role to the number 1 in ordinary arithmetic. Example: The identity matrix I2 (2x2 identity matrix) is given by: [1 0; 0 1]
  • The product of a matrix A and its inverse A^-1 is the identity matrix: A * A^-1 = A^-1 * A = I. Example: Let A = [2 1; 3 4] and A^-1 = [2 -1; -3 2]. Now, let’s verify the property A * A^-1 = I: A * A^-1 = [2 1; 3 4] * [2 -1; -3 2] = [1 0; 0 1] = I Therefore, A * A^-1 = I. Equation: To find the inverse of a matrix A, we can use the equation: A^-1 = (1/det(A)) * adj(A) Where adj(A) represents the adjugate of matrix A. Note: If the determinant of a matrix is zero, the matrix does not have an inverse.

Matrix and Determinant - Example Problems

Problem 1: Find the inverse of matrix M = [2 4; 6 8]. Solution: det(M) = 28 - 46 = 16 - 24 = -8 (non-zero) adj(M) = [8 -4; -6 2] M^-1 = (1/-8) * [8 -4; -6 2] = [-1/2 1/4; 3/4 -1/8] Therefore, the inverse of matrix M is M^-1 = [-1/2 1/4; 3/4 -1/8]. Problem 2: Verify that matrix N = [5 2; 1 3] is its own inverse. Solution: N * N = [5 2; 1 3] * [5 2; 1 3] = [27 14; 6 7] Since N * N is not equal to the identity matrix, matrix N is not its own inverse.

  1. Properties of Inverse (continued):
  • If A is an invertible matrix, then its inverse A^-1 is unique.
  • If A is an invertible matrix, then the product of A and its inverse A^-1 is the identity matrix: A * A^-1 = A^-1 * A = I.
  • A matrix A does not necessarily have an inverse if the product A * B = B * A = I, where B is a different matrix. Example: Consider the matrix X = [2 3; 1 2] and its inverse X^-1 = [2 -3; -1 2]. Let’s verify the property A * A^-1 = I: X * X^-1 = [2 3; 1 2] * [2 -3; -1 2] = [1 0; 0 1] = I Therefore, X * X^-1 = I.
  1. The Inverse of a 2x2 Matrix:
  • To find the inverse of a 2x2 matrix A = [a b; c d], we can use the following formula: A^-1 = (1/det(A)) * [d -b; -c a] where det(A) = ad - bc. Example: Let’s find the inverse of matrix Y = [4 3; 2 1]: Y = [4 3; 2 1] det(Y) = 41 - 32 = -2 (non-zero) Y^-1 = (1/-2) * [1 -3; -2 4] = [-1/2 3/2; 1 -2] Therefore, the inverse of matrix Y is Y^-1 = [-1/2 3/2; 1 -2].
  1. Identity Matrix:
  • The identity matrix is a square matrix in which all the diagonal elements are 1 and all other elements are 0.
  • The identity matrix, denoted by I, has the property that any matrix multiplied by I will equal the original matrix. Example: The identity matrix I3 (3x3 identity matrix) is given by: [1 0 0; 0 1 0; 0 0 1] Here are some properties of the identity matrix:
  • When an identity matrix is multiplied by a matrix A, the result is the matrix A: I * A = A.
  • When a matrix A is multiplied by an identity matrix, the result is the matrix A: A * I = A.
  1. Determinant of the Identity Matrix:
  • The determinant of the identity matrix is always equal to 1.
  • For any square matrix A, if det(A) = 1, then A is called a unimodular matrix. Example: Let’s calculate the determinant of the 3x3 identity matrix I3: det(I3) = 111 = 1 Therefore, the determinant of the identity matrix I3 is 1.
  1. Non-Invertible Matrices:
  • If a square matrix A does not have an inverse, it is called a singular matrix or a non-invertible matrix.
  • A matrix is singular if and only if its determinant is equal to zero: det(A) = 0. Example: Consider the matrix Z = [2 4; 6 12]. det(Z) = 212 - 46 = 24 - 24 = 0 Since the determinant is zero, matrix Z does not have an inverse.
  1. Matrices with Zero Determinant:
  • If the determinant of a matrix A is zero (det(A) = 0), then A does not have an inverse.
  • Matrices with zero determinants are not one-to-one mappings, and they have non-zero null space. Example: Consider the matrix W = [1 2 3; 2 4 6; 3 6 9]. det(W) = 1*(49 - 66) - 2*(29 - 63) + 3*(26 - 43) = 0 Since the determinant is zero, matrix W does not have an inverse.
  1. Special Matrices:
  • Diagonal matrix: A diagonal matrix is a square matrix in which all the elements outside the main diagonal are zero.
  • Scalar matrix: A scalar matrix is a diagonal matrix in which all the diagonal elements are equal. Example: Consider the following special matrices:
  • Diagonal matrix D1 = [2 0; 0 3]
  • Scalar matrix S = [5 0; 0 5] The inverse of a diagonal matrix or a scalar matrix exists if none of the diagonal elements are zero.
  1. Zero Matrix:
  • The zero matrix, denoted by O, is a matrix in which all the elements are zero. Properties of the zero matrix:
  • If A is any matrix, then A + O = O + A = A.
  • If A is any matrix, then A - O = A.
  • If A is any matrix, then A * O = O * A = O. Example: Consider a 2x2 zero matrix O = [0 0; 0 0]. Let’s perform some operations with the zero matrix:
  • [3 4; 1 2] + O = [3 4; 1 2]
  • [3 4; 1 2] - O = [3 4; 1 2]
  • [3 4; 1 2] * O = O
  1. Cofactor Matrix:
  • The cofactor matrix of a given square matrix A is obtained by taking the cofactors of the elements of A and arranging them in a matrix format. The cofactor of an element A(i,j) in a matrix A is given by: C(i,j) = (-1)^(i+j) * M(i,j) where M(i,j) is the minor of A(i,j). Example: Consider the matrix A = [2 3; 4 5]. The cofactor matrix C of matrix A is given by: C = [C(1,1) C(1,2); C(2,1) C(2,2)]
  1. Cofactor Matrix (continued):
  • The cofactor matrix C can also be obtained by finding the adjugate of A and dividing it by the determinant of A. Example: Consider the matrix A = [2 3; 4 5]. The adjugate matrix adj(A) is given by: adj(A) = [5 -3; -4 2] The cofactor matrix C of matrix A is obtained by dividing adj(A) by the determinant of A: C = (1/det(A)) * adj(A) = (1/(25 - 43)) * [5 -3; -4 2] = (1/2) * [5 -3; -4 2] = [5/2 -3/2; -2 1] Therefore, the cofactor matrix C of matrix A is C = [5/2 -3/2; -2 1].
  1. Applications of Inverse Matrices:
  • Inverse matrices have various applications in different fields, including:
  1. Solving Systems of Equations:
    • Inverse matrices can be used to solve systems of linear equations efficiently.
    • We can represent the system of equations as a matrix equation AX = B, where A is the coefficient matrix, X is the vector of variables, and B is the constant vector.
    • By multiplying both sides of the equation by the inverse of matrix A, we can solve for X: X = A^-1 * B.
  1. Finding the Solution to Linear Equations:
    • Inverse matrices help in finding the solution to linear equations.
    • If A is an invertible matrix, then the linear system AX = B has a unique solution, given by X = A^-1 * B.
  1. Computing Determinants:
    • Inverse matrices can be used to compute determinants of matrices.
    • If A is an invertible matrix, then the determinant of A is given by det(A) = 1/det(A^-1).
  1. Finding Solutions in Physics:
    • Inverse matrices are used in solving problems in physics, such as calculating electric current distribution in an electrical circuit or determining the motion of objects in dynamics.
  1. Image and Signal Processing:
    • Inverse matrices are used in image and signal processing techniques, such as image and audio compression.
    • They help in reconstructing the original image/signal from the compressed version using the inverse transformation.
  1. Solving a System of Linear Equations:
  • Inverse matrices provide a method for solving systems of linear equations.
  • Let’s consider a system of two linear equations: Equation 1: 3x + 2y = 7 Equation 2: 4x - y = 8 We can represent this system using a matrix equation AX = B, where: A = [3 2; 4 -1], X = [x; y], B = [7; 8] To solve for X, we can use the equation X = A^-1 * B.
  1. Solving a System of Linear Equations (continued): Let’s find the inverse of matrix A: A = [3 2; 4 -1] det(A) = 3*-1 - 4*2 = -11 (non-zero) adj(A) = [-1 -2; -4 3] A^-1 = (1/-11) * [-1 -2; -4 3] = [1/11 2/11; 4/11 -3/11] Now, let’s find the solution X = A^-1 * B: B = [7; 8] X = [1/11 2/11; 4/11 -3/11] * [7; 8] = [1; 2] Therefore, the solution to the system of linear equations is x = 1 and y = 2.
  1. Matrix Inverse and Square Matrices:
  • Some properties of matrix inverses for square matrices:
  1. If A is a square matrix and B is its inverse, then B is also a square matrix.
  1. If A and B are invertible matrices, then (AB)^-1 = B^-1 * A^-1.
  1. If A is an invertible matrix, then (A^-1)^-1 = A.
  1. If A is an invertible matrix, then (kA)^-1 = 1/k * A^-1, where k is a non-zero scalar. Example: Consider matrices P = [2 3; 1 4] and Q = [1 2; 3 5]. Let’s find the inverse of P and Q, and verify some of the properties discussed:
  • Inverse of P: P = [2 3; 1 4] det(P) = 24 - 31 = 5 (non-zero) P^-1 = (1/5) * [4 -3; -1 2] = [4/5 -3/5; -1/5 2/5]
  • Inverse of Q: Q = [1 2; 3 5