Matrices - Invertibility of Matrix

  • Today’s topic: Invertibility of Matrix
  • An invertible matrix is a square matrix that has an inverse
  • A matrix A is invertible if there exists a matrix B such that AB = BA = I, where I is the identity matrix

Conditions for Invertibility

  1. Matrix has to be a square matrix
  1. Determinant of the matrix should not be zero
  1. Matrix should have linearly independent columns or rows

Example: Invertible Matrix

Let’s consider a 3x3 matrix A: A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

  • A is a square matrix (3x3)
  • Determinant of A is non-zero (det(A) ≠ 0)
  • Columns of A are linearly independent

Example: Non-Invertible Matrix

Consider a 2x2 matrix B: B = [[2, 4], [1, 2]]

  • B is a square matrix (2x2)
  • Determinant of B is zero (det(B) = 0)
  • Columns of B are linearly dependent

Properties of Invertible Matrix

  1. Inverse of an invertible matrix is unique
  1. If A is invertible, then A^(-1) is also invertible and (A^(-1))^(-1) = A
  1. If A and B are invertible matrices, then AB is also invertible and (AB)^(-1) = B^(-1) * A^(-1)
  1. If A is an invertible matrix, then (A^T)^(-1) = (A^(-1))^T

Example: Finding Inverse of a Matrix

Let’s find the inverse of matrix A: A = [[1, 2], [3, 4]]

  • To find A^(-1), we will first compute the determinant of A:
    • det(A) = 14 - 23 = -2
  • Next, we will find the adjoint of A by interchanging the elements across the main diagonal: adj(A) = [[4, -2], [-3, 1]]
  • Finally, we can find A^(-1) by dividing the adjoint by the determinant: A^(-1) = (1/-2) * [[4, -2], [-3, 1]] = [[-2, 1], [3/2, -1/2]]

Properties of Invertible Matrix (Continued)

  1. If A is an invertible matrix, then the matrix obtained by multiplying any row (or column) of A by a non-zero scalar k is also invertible
  1. If A is an invertible matrix, then the matrix obtained by adding a multiple of any row (or column) of A to another row (or column) is also invertible
  1. If A and B are invertible matrices of the same order, then the matrix A * B is invertible, and its inverse is given by B^(-1) * A^(-1)
  1. The product of two invertible matrices is also invertible, but the product of two non-invertible matrices may or may not be invertible
  1. If A and B are invertible matrices, then (A^(-1))^(-1) = A and (AB)^(-1) = B^(-1) * A^(-1)

Example: Product of Two Invertible Matrices

Let’s consider two invertible matrices: A = [[2, 4], [1, 3]] B = [[5, 6], [7, 8]]

  • A is an invertible matrix because its determinant is non-zero
  • B is also an invertible matrix because its determinant is non-zero

Example: Product of Two Invertible Matrices (Continued)

We can find the product of A and B: A * B = [[2, 4], [1, 3]] * [[5, 6], [7, 8]] = [[2*5 + 4*7, 2*6 + 4*8], [1*5 + 3*7, 1*6 + 3*8]] = [[38, 52], [26, 36]]

  • The resulting matrix AB is also invertible

Example: Inverse of a Product of Two Matrices

Let’s find the inverse of the matrix AB: (AB)^(-1) = [[38, 52], [26, 36]]^(-1)

  • We can find the inverse of AB by finding the inverse of each individual matrix and multiplying them in reverse order

Example: Inverse of a Product of Two Matrices (Continued)

To find the inverse of AB, we can find the inverse of A and B: A^(-1) = [[2, 4], [1, 3]]^(-1) B^(-1) = [[5, 6], [7, 8]]^(-1)

  • We can use the previously discussed method to find the inverse of A and B

Example: Inverse of a Product of Two Matrices (Continued)

Let’s find the inverse of A: A^(-1) = (1/2) * [[3, -4], [-1, 2]]

  • Similarly, let’s find the inverse of B: B^(-1) = (1/10) * [[8, -6], [-7, 5]]

Example: Inverse of a Product of Two Matrices (Continued)

Now, we can find the inverse of AB by multiplying the inverses of A and B in reverse order: (AB)^(-1) = [[38, 52], [26, 36]]^(-1) = [[38, 52], [26, 36]]^(-1) = (1/10) * [[36, -52], [-26, 38]]

Invertibility of Identity Matrix

  • The identity matrix I is an invertible matrix
  • Its inverse is the same matrix I Example: I = [[1, 0], [0, 1]]
  • I is the identity matrix
  • I * I = I
  • I^(-1) = I

Singular Matrix

  • A square matrix that is not invertible is called a singular matrix or a non-invertible matrix
  • A singular matrix has a determinant equal to zero Example: S = [[2, 4], [3, 6]]
  • The determinant of S is zero

Properties of Singular Matrix

  1. A singular matrix does not have an inverse
  1. If A is a singular matrix, then kA is also a singular matrix for any non-zero scalar k
  1. If A and B are singular matrices of the same order, then AB is also a singular matrix
  1. If A is a non-singular matrix and k ≠ 0, then kA is a non-singular matrix
  1. The sum or difference of two non-singular matrices may or may not be non-singular

Example: Inverse of a Singular Matrix

Let’s find the inverse of a singular matrix S: S = [[2, 4], [3, 6]]

  • To find the inverse of S, we first compute the determinant of S:
    • det(S) = 26 - 43 = 0
  • Since the determinant is zero, S does not have an inverse.

Finding Inverse using Elementary Row Operations

  • The inverse of a matrix can also be found using elementary row operations
  • The row operation is performed on the given matrix A, and the same row operation is applied to the identity matrix I
  • The final form of matrix A should be the identity matrix, and the final form of the identity matrix should be the inverse of A

Example: Finding Inverse using Row Operations

Let’s find the inverse of matrix A using row operations: A = [[1, 2], [3, 7]]

  • We start with the given matrix A and the identity matrix: [A | I] = [[1, 2 | 1, 0], [3, 7 | 0, 1]]
  • Using row operations, we can transform the matrix A into the identity matrix:
    [R2 = R2 - 3R1] [A | I] = [[1, 2 | 1, 0], [0, 1 | -3, 1]] [R1 = R1 - 2R2] [A | I] = [[1, 0 | 7, -2], [0, 1 | -3, 1]]
  • The final matrix is the inverse of A: A^(-1) = [[7, -2], [-3, 1]]

Non-Square Matrices

  • Invertibility is applicable only to square matrices
  • Non-square matrices do not have an inverse

Example: Non-Invertible Non-Square Matrix

Consider a 2x3 matrix B: B = [[2, 4, 6], [1, 2, 3]]

  • B is a non-square matrix (2x3)
  • Since B is not a square matrix, it does not have an inverse

Invertibility and Linear Systems

  • Invertible matrices play a crucial role in solving linear systems of equations
  • An n x n matrix A is invertible if and only if the system of equations Ax = b has a unique solution for every vector b
  • In this case, the unique solution is given by x = A^(-1)b

Example: Solving Linear System using Invertible Matrix

Let’s consider a linear system of equations: ``

2x + 3y = 8

5x + 4y = 13 ``

  • We can represent the linear system in matrix form as Ax = b: A = [[2, 3], [5, 4]] x = [[x], [y]] b = [[8], [13]]
  • Since the matrix A is invertible, we can find the unique solution using x = A^(-1)b.

Cramer’s Rule

  • Cramer’s Rule provides a method to solve a linear system of equations using determinants
  • The solutions for the variables in the system can be found by evaluating determinants
  • The determinant of the coefficient matrix A is denoted as det(A)

Cramer’s Rule for a 2x2 Linear System

Consider a linear system of equations: ax + by = p cx + dy = q

  • The determinant of the coefficient matrix A is given by: det(A) = ad - bc
  • The solutions for the variables x and y can be found using Cramer’s Rule: x = (pd - bq) / det(A) y = (aq - pc) / det(A)

Example: Solving Linear System using Cramer’s Rule

Let’s solve the following linear system of equations using Cramer’s Rule: ``

2x + 3y = 8

5x + 4y = 13 ``

  • The coefficient matrix A is: A = [[2, 3], [5, 4]]
  • The determinant of A is: det(A) = 2*4 - 3*5 = -7
  • Using Cramer’s Rule, we can find the values of x and y: x = (8*4 - 3*13) / (-7) = 3 y = (2*13 - 8*5) / (-7) = -1
  • Hence, the solution to the linear system is: x = 3, y = -1