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2.2 inverse of a matrix

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This document discusses inverses of matrices. It defines an invertible matrix as a square matrix A that has an inverse matrix B such that AB and BA are the identity matrix. It also defines singular and non-singular matrices. Theorems are provided to determine if a 2x2 matrix is invertible based on its determinant, and to solve systems of equations using the inverse matrix. Elementary matrices from row operations on the identity matrix are introduced. An algorithm for finding the inverse of an invertible matrix using row operations on the augmented matrix [A|I] is also given.

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2.2 Inverse of a Matrix
Definition [Invertible Matrix] An nxn matrix A is said to be invertible if there is an nxn matrix B such that AB=BA=I in which case B is an inverse of A and we write B=A-1
Definition [Singular and Non Singular Matrices] A matrix that is not invertible is called a singular matrix. A matrix that is invertible is called non singular matrix.
Example: The matrix B= 2 1 1.5 0.5 is an inverse of A = 1 2 3 4 . Solution:
Remark: Statement: If a square matrix has an inverse then it is unique. proof: Let A be an invertible matrix. If B be the inverse of the matrix A then by definition, AB = BA = I (i) Suppose if possible, let C be another inverse of A, then again by definition, AC = CA = I.(ii) Now, C = CI = CAB from (i) = IB from (ii) = B, thus B is unique.
Theorem 4 Let A= , ad bc 0, then A is invertible and 癌1 = 1 . If ad-bc=0, then A is invertible. Proof:
2.2 inverse of a matrix
Definition [Determinant] Let A= . The quantity ad-bc is called the determinant of A. It is denoted by detA or or . Thus, =ad-bc
Note: 1. Theorem 4 guarantees that a 2x2 matrix A is invertible if and only if detA 0. 2. A matrix A is singular if detA=0 and it is non singular if detA 0.
Example: Find the inverse of A= 2 4 3 5 . Solution:
Theorem 5 If A is an invertible nxn matrix, then for each b in Rn, the equation Ax=b has the unique solution x=A-1b. Proof:
Example: Use the inverse of the matrix to solve the system, 2x + 4y = 6 , 3x + 5y = 8. Solution:
Theorem 6 a. If A is an invertible matrix, then A-1 is invertible and (A-1)-1=A. b. If A and B are nxn invertible matrices, then so is AB and the inverse of AB is the product of the inverses of A and B the reverse order. That is (AB)-1=B-1A-1. c. If A is an invertible matrix, then so is AT, and the inverse of AT is the transpose of A-1. That is (AT)-1=(A-1)T. Proof:
2.2 inverse of a matrix
Elementary Matrices A matrix that is obtained by performing a single elementary row operation in an identity matrix is called an elementary matrix. e.g. 1 = 1 0 0 0 1 0 4 0 1 , is an elementary matrix which is obtained by performing the operation R3R3 4R1
Algorithm for finding A-1 Let A be and invertible matrix. 1. Row reduce the augmented matrix [A | I]. 2. If A is row equivalent to I, then [A | I] is row equivalent to [I | A-1] i.e. [A | I] ~ ~ [I | A-1]. 3. Otherwise , A does not have an inverse.
Example

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2.2 inverse of a matrix

  • 1. 2.2 Inverse of a Matrix
  • 2. Definition [Invertible Matrix] An nxn matrix A is said to be invertible if there is an nxn matrix B such that AB=BA=I in which case B is an inverse of A and we write B=A-1
  • 3. Definition [Singular and Non Singular Matrices] A matrix that is not invertible is called a singular matrix. A matrix that is invertible is called non singular matrix.
  • 4. Example: The matrix B= 2 1 1.5 0.5 is an inverse of A = 1 2 3 4 . Solution:
  • 5. Remark: Statement: If a square matrix has an inverse then it is unique. proof: Let A be an invertible matrix. If B be the inverse of the matrix A then by definition, AB = BA = I (i) Suppose if possible, let C be another inverse of A, then again by definition, AC = CA = I.(ii) Now, C = CI = CAB from (i) = IB from (ii) = B, thus B is unique.
  • 6. Theorem 4 Let A= , ad bc 0, then A is invertible and 癌1 = 1 . If ad-bc=0, then A is invertible. Proof:
  • 8. Definition [Determinant] Let A= . The quantity ad-bc is called the determinant of A. It is denoted by detA or or . Thus, =ad-bc
  • 9. Note: 1. Theorem 4 guarantees that a 2x2 matrix A is invertible if and only if detA 0. 2. A matrix A is singular if detA=0 and it is non singular if detA 0.
  • 10. Example: Find the inverse of A= 2 4 3 5 . Solution:
  • 11. Theorem 5 If A is an invertible nxn matrix, then for each b in Rn, the equation Ax=b has the unique solution x=A-1b. Proof:
  • 12. Example: Use the inverse of the matrix to solve the system, 2x + 4y = 6 , 3x + 5y = 8. Solution:
  • 13. Theorem 6 a. If A is an invertible matrix, then A-1 is invertible and (A-1)-1=A. b. If A and B are nxn invertible matrices, then so is AB and the inverse of AB is the product of the inverses of A and B the reverse order. That is (AB)-1=B-1A-1. c. If A is an invertible matrix, then so is AT, and the inverse of AT is the transpose of A-1. That is (AT)-1=(A-1)T. Proof:
  • 15. Elementary Matrices A matrix that is obtained by performing a single elementary row operation in an identity matrix is called an elementary matrix. e.g. 1 = 1 0 0 0 1 0 4 0 1 , is an elementary matrix which is obtained by performing the operation R3R3 4R1
  • 16. Algorithm for finding A-1 Let A be and invertible matrix. 1. Row reduce the augmented matrix [A | I]. 2. If A is row equivalent to I, then [A | I] is row equivalent to [I | A-1] i.e. [A | I] ~ ~ [I | A-1]. 3. Otherwise , A does not have an inverse.