965 9.8 Matrix Inverses 9.8 Matrix Inverses ■ Identity Matrices ■ Multiplicative Inverses ■ Solution of Systems Using Inverse Matrices We have seen several parallels between the set of real numbers and the set of matrices. Another similarity is that both sets have identity and inverse elements for multiplication. Identity Matrices By the identity property for real numbers, a # 1 = a and 1 # a = a for any real number a. If there is to be a multiplicative identity matrix I, such that AI = A and IA = A, for any matrix A, then A and I must be square matrices of the same dimension. 2 : 2 Identity Matrix I2 represents the 2 * 2 identity matrix. I2 = c 1 0 0 1d To verify that I2 is the 2 * 2 identity matrix, we must show that AI = A and IA = A for any 2 * 2 matrix A. Let A = c x z y wd . Then AI = c x z y wd c 1 0 0 1d = c x # 1 + y # 0 z # 1 + w # 0 x # 0 + y # 1 z # 0 + w # 1d = c x z y wd = A, and IA = c 1 0 0 1d c x z y wd = c 1 # x + 0 # z 0 # x + 1 # z 1 # y + 0 # w 0 # y + 1 # wd = c x z y wd = A. Generalizing, there is an n * n identity matrix for every n * n square matrix. The n * n identity matrix has 1s on the main diagonal and 0s elsewhere. n : n Identity Matrix The n * n identity matrix is In. In = ≥ 1 0 P 0 0 1 P 0 O O aij O 0 0 P 1 ¥ The element aij = 1 when i = j (the diagonal elements), and aij = 0 otherwise.
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