Tiêu đề Inverse of a matrix.pdf ✅

Inverse of a Matrix Page | 7 Definition 8.2.1 An nൈn matrix A is called invertible (or nonsingular) if there exists an nൈn matrix B such that AB ൌ BA ൌ I௡. ሺ1ሻ Here I௡ is the identity matrix of order n which has 1’s in the main diagonal and 0’s everywhere else, namely I௡ ൌ ൦ 1 0 ⋮ 0 0 1 ⋮ 0 ⋯ ⋯ ⋮ ⋯ 0 0 ⋮ 1 ൪. If the order of I௡ is clear from the context, we will simply write I for I௡. Here B is called an inverse of A. Since the eq. (1) is symmetric in A and B, we can also say A is an inverse of B. For example, A ൌ ൥ 1 0 െ2 2 1 0 3 2 1 ൩ is invertible since there exists a matrix B ൌ ൥ െ1 4 െ2 2 െ7 4 െ1 2 െ1 ൩ such that AB ൌ ൥ 1 0 െ2 2 1 0 3 2 1 ൩ ൥െ1 4 െ2 2 െ7 4 െ1 2 െ1 ൩ ൌ ൥ 100 010 001 ൩ ൌ I, BA ൌ ൥ െ1 4 െ2 2 െ7 4 െ1 2 െ1 ൩ ൥1 0 െ2 2 1 0 3 2 1 ൩ ൌ ൥ 100 010 001 ൩ ൌ I. Not every square matrix is invertible. If A is not an invertible matrix, then we say A is noninvertible (or singular). For example, the zero matrix O ൌ ൥ 000 000 000 ൩ is noninvertible. Clearly, there cannot exist any matrix B such that OB ൌ BO ൌ I since OB ൌ BO ൌ O.
Inverse of a Matrix Page | 8 Theorem (Uniqueness of an Inverse) If A is an invertible matrix, then its inverse is unique. Proof. Let A be an nൈn invertible matrix. So, there exists an nൈn matrix B such that AB ൌ BA ൌ I. Let us assume that there exists another nൈn matrix C such that AC ൌ CA ൌ I. To show that BൌC, we use B ൌ BI [Definition of Identity matrix] ൌ BሺACሻ [C is an inverse of A] ൌ ሺBAሻC [Associative Property for Matrix Multiplication] ൌ IC [B is an inverse of A] ൌ C [Definition of Identity matrix] The proof is complete. ∎ Notation: If A is invertible and since its inverse is unique, we will use the symbol Aିଵ to denote the inverse of A.
Properties of Inverse of a Matrix Page | 9 Theorem If A be an nൈn invertible matrix, then the following are true. 1. ሺAିଵሻିଵ ൌ A. 2. ሺA௞ሻିଵ ൌ AିଵAିଵ ⋯ A ᇣᇧᇧᇧᇤᇧᇧᇧିଵ ᇥ ௞ ୫ୟ୬୷ ୤ୟୡ୲୭୰ୱ ൌ ሺAିଵሻ௞ 3. ሺcAሻିଵ ൌ cିଵAିଵ if c ് 0. 4. ሺA்ሻିଵ ൌ ሺAିଵሻ் Proof. 1. By definition, AିଵሺAିଵሻିଵ ൌ I Now by multiplying both sides by A from the left, we get the result as follows AሺAିଵሺAିଵሻିଵሻ ൌ AI ሺAAିଵሻሺAିଵሻିଵ ൌ A IሺAିଵሻିଵ ൌ A ሺAିଵሻିଵ ൌ A 2. By definition, A௞ሺA௞ሻିଵ ൌ I Since k is a positive integer, ሺ ᇣAAᇧᇧᇤ...ᇧᇧAᇥሻ ௞ ሺA௞ሻିଵ ൌ I Now multiplying both sides by Aିଵ from the left by k many times we obtain the desired result. ሺA௞ሻିଵ ൌ ሺAିଵAିଵ ... Aିଵ ᇣᇧᇧᇧᇧᇤᇧᇧᇧᇧᇥሻ ௞ ൌ ሺAିଵሻ௞ 3. By definition, ሺcAሻሺcAሻିଵ ൌ I AሺcAሻିଵ ൌ cିଵI
Properties of Inverse of a Matrix Page | 10 ሺcAሻିଵ ൌ AିଵሺcିଵIሻ ൌ cିଵሺAିଵIሻ ൌ cିଵAିଵ 4. By definition, AAିଵ ൌ I Taking the transpose of both sides, we obtain ሺAAିଵሻ் ൌ I் ሺAିଵሻ்A் ൌ I Here we have used ሺABሻ் ൌ B்A் and I் ൌ I. ሺAିଵሻ்A் ൌ I ∴ ሺA்ሻିଵ ൌ ሺAିଵሻ். Theorem. If A and B are two nൈn invertible matrices, then AB is invertible and ሺABሻିଵ ൌ BିଵAିଵ. (This is known as Socks‐shoes property. You might be surprised to see that taking the multiplicative inverse reverses the order of multiplication. So interpret A as putting on socks, and B as putting on shoes. To reverse the operation AB of putting on both socks and shoes, you must reverse the order: you take off shoes first, then the socks, and so the inverse operation is BିଵAିଵ.) Proof. The proof is straightforward. Observe that ሺBିଵAିଵሻሺABሻ ൌ ൫BିଵሺAିଵAሻ൯B ൌ ሺBିଵIሻB ൌ BିଵB ൌ I