4.5 Inverse of a Square Matrix For any real non-zero number a there exists a unique non-zero real 1 number b such that a · b = b · a = 1. Clearly, b = a, and b is called multiplicative inverse (or reciprocal) of a. For example, Similarly, we can consider multiplicative inverse for matrices. It may not exists for any non-zero matrix. However, if it does exist then it is unique. First, we need to define multiplicative identity (for real numbers, multiplicative inverse is 1). Identity Matrix for Multiplication Definition (Identity Matrix) The identity element for multiplication for the set of all square matrices of order n is the square matrix of order n, denoted by I, with 1's along the principal diagonal (from the upper left corner to the lower right) and 0's elsewhere. For example, Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix Properties of Identity Matrix For any square n × n matrix M, we have MI = IM = M; where I is n × n identity matrix. For example, 2 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix Inverse of a Square Matrix Definition (Inverse of a Square Matrix) Let M be a square matrix of order n and I be the identity matrix of order n. If there exists a matrix M −1 (read M inverse) such that M −1M = MM −1 = I then M −1 is called the multiplicative inverse of M or, more simply, the inverse of M. If no such matrix exists, then M is said to be a singular matrix. Note: M −1 and M commute (M −1M = MM −1): Note: inverse is unique. That is, if A and B are both inverse of M, then A = B. 2 −6 If M = , we can find M −1 using the definitions of matrix 1 2 equality, matrix multiplication and the inverse: 3 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix 4 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix As mentioned before, inverse may not exist for a non-zero matrix. For 2 4 example, if N = 3 6 5 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix 2 3 −1 13 Let's try a 3 × 3 matrix M = 4−1 1 05 1 0 1 6 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix 7 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix Theorem 1 (Inverse of a Square Matrix M) If [M j I] is transformed by row operations into [I j B], then the resulting matrix B is M −1. However, if we obtain all 0s in one or more rows to the left of the vertical line, then M −1 does not exist. 2 −6 For example, if M = 1 2 8 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix 21 2 33 Consider M = 40 0 05 4 5 6 20 1 23 Consider M = 40 3 45 0 5 6 9 Ch 4. Linear Systems and Matrices 4.5 Inverse of a Square Matrix a b Consider an arbitrary 2 × 2 matrix M = c d 10.