INVERSE OF MATRIX
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The document discusses matrix inverses and how to find them using elementary row operations. It defines what an inverse matrix is and provides examples. It then presents an algorithm for finding the inverse of a 3x3 matrix using 6 steps of elementary row operations to convert the matrix to the identity matrix. A similar 2-step algorithm is given for a 2x2 matrix. The document emphasizes that the inverse of a square matrix, if it exists, is unique.
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Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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INVERSE OF MATRIX
- 3. 3 If square matrices A and B such that AB = BA = I, then B is called the inverse of A (symbol: A-1); and A is called the inverse of B (symbol: B-1). The inverse of a matrix or invertible matrix 6 2 3 1 1 0 1 0 1 B − − = − − Show B is the inverse of matrix A. 1 2 3 1 3 3 1 2 4 A = Example: 1 0 0 0 1 0 0 0 1 AB BA = = Ans: Note that Can you show the details?
- 4. Theorem :if a square matrix has an inverse, then it is unique. Proof: Let A be a square matrix of order n which is invertible, let B, C (square matrix of order n) be two inverse of A, then (by definition) AB=𝐼 𝑛 = 𝐵𝐴 ………(i) AC=𝐼 𝑛=CA………(ii) Now AB=𝐼 𝑛 C(AB)=C𝐼 𝑛 [multiply both side by matrix C] (CA)B=C [using Associative laws A(BC)=(AB)C] 𝐼 𝑛B=C [using (ii) B=C Therefore, inverse of a square matrix, if it exist, is unique. 4
- 5. Example : 5
- 6. Inverse of matrix by elementary operations 6 Convert to I using elementary row operations
- 7. Important steps to find inverse of 3x3 order matrix using elementary operations 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 If we follows these following steps then we can get answer is very short ways. Step-1 make it 1 Step-2 make to 𝑎21 𝑎𝑛𝑑 𝑎31 0 Step-3 make it 1 Step-4 make them 0 Step-5 make it 1 Step-6 make them 0
- 8. Algorithm for 3 x 3 order matrix: Step-1 make 𝑎11=1 by operation 𝑅1 → 1 𝑎11 𝑅1(𝑎11 ≠ 0). If 𝑎11=0 then 𝑅1 ↔ 𝑅2 𝑜𝑟𝑅3 then apply𝑅1 → 1 𝑎11 𝑅1. Step-2 Apply 𝑅2 → 𝑅2 −(𝑎21) 𝑅1 and 𝑅3 → 𝑅3 −(𝑎31) 𝑅1to obtain 𝑎21 = 𝑎31 = 0 Step-3 apply 𝑅2 → 1 𝑎22 𝑅2(𝑖𝑓 𝑎22 ≠ 0) or 𝑅2 ↔ 𝑅3 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑅2 → 1 𝑎22 𝑅2(if need be) to make 𝑎22 = 1 Step-4Apply 𝑅1 → 𝑅1 −(𝑎12) 𝑅2 and 𝑅3 → 𝑅3 −(𝑎32) 𝑅2to obtain 𝑎12 = 𝑎32 = 0 Step-5 Apply 𝑅3 → 1 𝑎33 𝑅3 𝑡𝑜 𝑚𝑎𝑘𝑒 𝑎33 = 1 Step-6 Apply 𝑅2 → 𝑅2 −(𝑎23) 𝑅3 and 𝑅1 → 𝑅1 −(𝑎13) 𝑅3to obtain 𝑎23 = 𝑎13 = 0
- 9. Algorithm for 2 x 2 order matrix: Step-1 make 𝑎11=1 by operation 𝑅1 → 1 𝑎11 𝑅1(𝑖𝑓 𝑎11 ≠ 0). If 𝑎11=0 then 𝑅1 ↔ 𝑅2 then apply𝑅1 → 1 𝑎11 𝑅1. Step-2 Apply 𝑅2 → 𝑅2 −(𝑎21) 𝑅1 to obtain 𝑎21 = 0 Step-3 apply 𝑅2 → 1 𝑎22 𝑅2(𝑖𝑓 𝑎22 ≠ 0) or 𝑅2 ↔ 𝑅3 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑅2 → 1 𝑎22 𝑅2(if need be) to make 𝑎22 = 1 Step-4Apply 𝑅1 → 𝑅1 −(𝑎12) 𝑅2 to obtain 𝑎12 = 0
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