Determinants - Ways of calculating inverses

  • Inverse of a matrix
  • Adjoint method
  • Row reduction method
  • Determinant method
  • Cofactor method

Inverse of a matrix

  • The inverse of a square matrix A is denoted as A^-1
  • If A * A^-1 = I, where I is the identity matrix, then A^-1 is the inverse of A
  • Inverse of a matrix exists only if its determinant is non-zero

Adjoint method

  • Let’s consider a square matrix A
  • The adjoint of A, denoted as adj(A), is obtained by taking the transpose of the matrix of cofactors of A
  • The formula for finding adj(A) is:
    • adj(A) = (C11 C21 … Cn1; C12 C22 … Cn2; …; C1n C2n … Cnn)

Row reduction method

  • Start with the given matrix A on the left and an identity matrix I on the right
  • Perform row operations on both matrices simultaneously to convert A into the identity matrix
  • The resulting matrix on the right will be the inverse of A Example: A = [2 1; 3 2] I = [1 0; 0 1] Perform row operations on both matrices until A becomes the identity matrix

Determinant method

  • Calculate the determinant of the given matrix A
  • If the determinant is non-zero, proceed to the next step
  • Find the adjoint matrix adj(A)
  • Multiply adj(A) by 1/det(A) to obtain the inverse of A Example: A = [2 1; 3 2] det(A) = 4 adj(A) = [2 -1; -3 2] A^-1 = (1/4) * [2 -1; -3 2]

Cofactor method

  • Calculate the determinant of the given matrix A
  • If the determinant is non-zero, proceed to the next step
  • Find the matrix of cofactors, which is obtained by taking the determinant of each minor matrix and applying alternating signs
  • Transpose the matrix of cofactors to obtain the adjoint of A
  • Multiply adj(A) by 1/det(A) to obtain the inverse of A Example: A = [2 1; 3 2] det(A) = 4 Matrix of cofactors: [2 -3; -1 2] adj(A) = [2 -1; -3 2] A^-1 = (1/4) * [2 -1; -3 2]

Summary

  • Inverse of a matrix is denoted as A^-1
  • Inverse of a matrix exists only if its determinant is non-zero
  • Ways of calculating inverses include:
    • Adjoint method
    • Row reduction method
    • Determinant method
    • Cofactor method

Practice Exercise

  1. Find the inverse of the matrix A = [3 1; 2 5] using the adjoint method.
  1. Use the row reduction method to find the inverse of the matrix B = [4 2; 1 3].
  1. Calculate the inverse of the matrix C = [2 3; 1 4] using the determinant method.
  1. Find the inverse of the matrix D = [1 5; 3 2] using the cofactor method.

Solutions

  1. A = [3 1; 2 5] det(A) = 13 adj(A) = [5 -1; -2 3] A^-1 = (1/13) * [5 -1; -2 3]
  1. B = [4 2; 1 3] det(B) = 10 adj(B) = [3 -2; -1 4] B^-1 = (1/10) * [3 -2; -1 4]
  1. C = [2 3; 1 4] det(C) = 5 adj(C) = [4 -3; -1 2] C^-1 = (1/5) * [4 -3; -1 2]
  1. D = [1 5; 3 2] det(D) = -13 adj(D) = [2 -5; -3 1] D^-1 = (-1/13) * [2 -5; -3 1]
  1. Adjoint Method
  • The adjoint of a square matrix A is obtained by taking the transpose of the matrix of cofactors of A
  • Formula: adj(A) = (C11 C21 … Cn1; C12 C22 … Cn2; …; C1n C2n … Cnn)
  • Example: Let A = [3 2; 1 4]. Calculate adj(A).
  1. Row Reduction Method
  • Start with the given matrix A on the left and an identity matrix I on the right
  • Perform row operations on both matrices to convert A into the identity matrix
  • The resulting matrix on the right will be the inverse of A
  • Example: Let A = [1 2; 3 4]. Use row reduction to find the inverse of A.
  1. Determinant Method
  • Calculate the determinant of the given matrix A
  • If the determinant is non-zero, proceed to the next step
  • Find the adjoint matrix adj(A)
  • Multiply adj(A) by 1/det(A) to obtain the inverse of A
  • Example: Let A = [2 3; 1 4]. Calculate det(A) and find the inverse of A using the determinant method.
  1. Cofactor Method
  • Calculate the determinant of the given matrix A
  • If the determinant is non-zero, proceed to the next step
  • Find the matrix of cofactors by taking the determinant of each minor matrix and applying alternating signs
  • Transpose the matrix of cofactors to obtain the adjoint of A
  • Multiply adj(A) by 1/det(A) to obtain the inverse of A
  • Example: Let A = [5 4; 3 2]. Calculate det(A) and find the inverse of A using the cofactor method.
  1. Practice Exercise
  1. Find the inverse of the matrix A = [2 1; 3 5] using the adjoint method.
  1. Use the row reduction method to find the inverse of the matrix B = [4 2; 1 6].
  1. Calculate the inverse of the matrix C = [3 2; 1 4] using the determinant method.
  1. Find the inverse of the matrix D = [1 4; 2 3] using the cofactor method.
  1. Solution 1
  • A = [2 1; 3 5]
  • det(A) = 7
  • adj(A) = [5 -1; -3 2]
  • A^-1 = (1/7) * [5 -1; -3 2]
  1. Solution 2
  • B = [4 2; 1 6]
  • det(B) = 22
  • adj(B) = [6 -2; -1 4]
  • B^-1 = (1/22) * [6 -2; -1 4]
  1. Solution 3
  • C = [3 2; 1 4]
  • det(C) = 10
  • adj(C) = [4 -2; -1 3]
  • C^-1 = (1/10) * [4 -2; -1 3]
  1. Solution 4
  • D = [1 4; 2 3]
  • det(D) = -5
  • adj(D) = [3 -4; -2 1]
  • D^-1 = (-1/5) * [3 -4; -2 1]
  1. Summary
  • Adjoint method involves taking the transpose of the matrix of cofactors
  • Row reduction method uses row operations to convert the given matrix into the identity matrix
  • Determinant method uses the determinant and adjoint of the matrix to find the inverse
  • Cofactor method uses the determinant of each minor matrix and the adjoint to find the inverse
  • Practice exercises and solutions help reinforce the concepts
  1. Determinant Method - Example
  • Let’s find the inverse of matrix A = [3 2; 1 4] using the determinant method
  • First, calculate the determinant of A: det(A) = (34) - (21) = 10
  • Next, find the adjoint matrix adj(A) = [4 -2; -1 3]
  • Finally, multiply adj(A) by 1/det(A) to get the inverse: A^-1 = (1/10) * [4 -2; -1 3]
  1. Cofactor Method - Example
  • Now, let’s use the cofactor method to find the inverse of matrix B = [5 4; 3 2]
  • Calculate the determinant of B: det(B) = (52) - (43) = -7
  • Find the matrix of cofactors: [-2 -3; 4 5]
  • Transpose the matrix of cofactors to get the adjoint: adj(B) = [-2 4; -3 5]
  • Multiply adj(B) by 1/det(B) to obtain the inverse: B^-1 = (-1/7) * [-2 4; -3 5]
  1. Practice Exercise - Solution 1
  • Let A = [2 1; 3 5]
  • det(A) = (25) - (13) = 7
  • adj(A) = [5 -1; -3 2]
  • A^-1 = (1/7) * [5 -1; -3 2]
  1. Practice Exercise - Solution 2
  • Let B = [4 2; 1 6]
  • det(B) = (46) - (21) = 22
  • adj(B) = [6 -2; -1 4]
  • B^-1 = (1/22) * [6 -2; -1 4]
  1. Practice Exercise - Solution 3
  • Let C = [3 2; 1 4]
  • det(C) = (34) - (21) = 10
  • adj(C) = [4 -2; -1 3]
  • C^-1 = (1/10) * [4 -2; -1 3]
  1. Practice Exercise - Solution 4
  • Let D = [1 4; 2 3]
  • det(D) = (13) - (42) = -5
  • adj(D) = [3 -4; -2 1]
  • D^-1 = (-1/5) * [3 -4; -2 1]
  1. Summary
  • The adjoint method involves taking the transpose of the matrix of cofactors
  • In the row reduction method, row operations are performed on both matrices
  • The determinant method uses the determinant and adjoint of the matrix
  • The cofactor method involves finding the matrix of cofactors and its transpose
  • Practice exercises and solutions have been provided for additional practice
  1. Review: Inverse of a Matrix
  • The inverse of a matrix A is denoted as A^-1
  • It exists only if the determinant of A is non-zero
  • Different methods for calculating inverses include:
    • Adjoint method: taking the transpose of the matrix of cofactors
    • Row reduction method: performing row operations on both matrices simultaneously
    • Determinant method: finding the adjoint and multiplying by 1/det(A)
    • Cofactor method: finding the transpose of the matrix of cofactors
  1. Tips for Finding Inverses
  • The row reduction method is useful when you already have a matrix in the form [A|I] where A is the given matrix and I is the identity matrix.
  • The determinant method is efficient when the determinant of the matrix is known or easy to compute.
  • The cofactor method is useful for finding the inverse of larger matrices where computing determinants may be time-consuming.
  • It’s essential to check your answers by multiplying the matrix and its inverse to verify if they indeed yield the identity matrix.
  1. Final Thoughts
  • Finding the inverse of a matrix is an important concept in linear algebra.
  • It allows us to solve systems of linear equations, compute the solutions of linear systems, and perform other mathematical operations.
  • Understanding the different methods for calculating inverses is crucial for success in studying matrices and related topics.
  • Remember to practice these techniques to become proficient in finding inverses of matrices.