SECTION 11.4 Matrix Algebra 803 Multiplication with an Identity Matrix Let = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ A B 1 2 0 0 1 3 and 3 2 4 6 5 2 Find: (a) AI3 (b) I A2 (c) BI2 Solution EXAMPLE 11 (a) = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = AI A 1 2 0 0 1 3 1 0 0 0 1 0 0 0 1 1 2 0 0 1 3 3 (b) = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = I A A 1 0 0 1 1 2 0 0 1 3 1 2 0 0 1 3 2 (c) BI B 3 2 4 6 5 2 1 0 0 1 3 2 4 6 5 2 2 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = For an n by n square matrix, the entries located in row i, column i, i n 1 , ≤ ≤ are called the diagonal entries or the main diagonal . The n by n square matrix whose diagonal entries are 1’s, and all other entries are 0’s, is called the identity matrix I .n For example, = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ I I 1 0 0 1 1 0 0 0 1 0 0 0 1 2 3 and so on. Example 11 demonstrates the following property: Identity Property • If A is an m by n matrix, then I A A AI A and m n = = • If A is an n by n square matrix, then AI I A A n n = = DEFINITION Inverse of a Matrix Let A be a square n by n matrix. If there exists an n by n matrix A 1− (read as “ A inverse”) for which AA A A In 1 1 = = − − then A 1− is called the inverse of the matrix A. An identity matrix has properties similar to those of the real number 1. In other words, the identity matrix is a multiplicative identity in matrix algebra. 4 Find the Inverse of a Matrix
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