Lecture 5: Matrix Operations: Inverse • Inverse of a matrix • Computation of inverse using co-factor matrix • Properties of the inverse of a matrix • Inverse of special matrices • Unit Matrix • Diagonal Matrix • Orthogonal Matrix • Lower/Upper Triangular Matrices 1 Matrix Inverse • Inverse of a matrix can only be defined for square matrices. • Inverse of a square matrix exists only if the determinant of that matrix is non-zero. • Inverse matrix of 퐴 is noted as 퐴−1. • 퐴퐴−1 = 퐴−1퐴 = 퐼 • Example: 2 −1 0 1 • 퐴 = , 퐴−1 = , 1 0 −1 2 2 −1 0 1 0 1 2 −1 1 0 • 퐴퐴−1 = = 퐴−1퐴 = = 1 0 −1 2 −1 2 1 0 0 1 2 Inverse of a 3 x 3 matrix (using cofactor matrix) • Calculating the inverse of a 3 × 3 matrix is: • Compute the matrix of minors for A. • Compute the cofactor matrix by alternating + and – signs. • Compute the adjugate matrix by taking a transpose of cofactor matrix. • Divide all elements in the adjugate matrix by determinant of matrix 퐴. 1 퐴−1 = 푎푑푗(퐴) det(퐴) 3 Inverse of a 3 x 3 matrix (using cofactor matrix) 3 0 2 퐴 = 2 0 −2 0 1 1 0 −2 2 −2 2 0 1 1 0 1 0 1 2 2 2 0 2 3 2 3 0 Matrix of Minors = = −2 3 3 1 1 0 1 0 1 0 2 3 2 3 0 0 −10 0 0 −2 2 −2 2 0 2 2 2 1 −1 1 2 −2 2 Cofactor of A (퐂) = −2 3 3 .∗ −1 1 −1 = 2 3 −3 0 −10 0 1 −1 1 0 10 0 2 2 0 adj A = CT = −2 3 10 2 −3 0 2 2 0 0.2 0.2 0 1 1 A-1 = ∗ adj A = −2 3 10 = −0.2 0.3 1 |퐴| 10 2 −3 0 0.2 −0.3 0 4 Properties of Inverse of a Matrix • (A-1)-1 = A • (AB)-1 = B-1A-1 • (kA)-1 = k-1A-1 where k is a non-zero scalar. • (AT)-1 = (A-1)T 5 Inverse of Identity matrices • Inverse of identity matrix is itself. • Because: • 퐼퐼 = 퐼 • Example: 1 2 1 0 1 2 • 퐴퐼 = 퐴, = 3 4 0 1 3 4 1 0 1 2 1 2 • 퐼−1퐴 = 퐴, = 0 1 3 4 3 4 Inverse of Diagonal matrices • The determinant of a diagonal matrix is the product of its diagonal elements. • If they all are non-zero, then determinant is non-zero and the matrix is invertible. • The inverse of a diagonal matrix A is another diagonal matrix B whose diagonal elements are the reciprocals of the diagonal elements of A. • Example: 2 0 0 • 퐴 = 0 1 0 , 퐴 = 2 × 1 × −1 = −2 (≠ 0) 0 0 −1 1 0 0 −1 2 • 퐴 = 0 1 0 0 0 −1 7 Inverse of Orthonormal matrices • Earlier, we saw that multiplication of an orthogonal (orthonormal) matrix in its transpose results in identity matrix • If 퐴 is an orthonormal matrix, its inverse is equal to its transpose • 퐴 is an orthonormal 푛 × 푛 matrix 푇 • Recall 퐴퐴 = 퐼푛 , where 퐼푛 is a 푛 × 푛 identity matrix • So, 퐴푇 = 퐴−1 8 Inverse of Upper/Lower Triangular Matrices • Inverse of an upper/lower triangular matrix is another upper/lower triangular matrix. • Inverse exists only if none of the diagonal element is zero. • Can be computed from first principles: Using the definition of an Inverse. 퐴퐴−1 = 퐼. No need to compute determinant. • Diagonal elements of 퐴−1is the reciprocal of the elements of 퐴. • Other elements are iteratively computed such that the product of the matrices is 퐼. Inverse of Upper/Lower Triangular Matrices Upper Triangular Matrix: 2 1 0.5 푋 2 1 0.5 푋 1 0 퐴 = ; 퐴−1 ; 0 −1 0 −1 0 −1 0 −1 0 1 푆표푙푣푖푛푔 푓표푟 푋 푤푒 푔푒푡 푋 = 0.5 Lower Triangular Matrix: 2 0 0.5 0 2 0 0.5 0 1 0 퐵 = ; 퐵−1 = ; = 2 1 푋 1 2 1 푋 1 0 1 푆표푙푣푖푛푔 푓표푟 푋 푤푒 푔푒푡 푋 = −1.