Download e-book for kindle: A University Algebra: An Introduction to Classic and Modern by D. E. Littlewood

By D. E. Littlewood

ISBN-10: 0486627152

ISBN-13: 9780486627151

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Example text

From its form, this determinant is both linear and alternating in the rows o f [asJ. Hence it is a multiple o f the deter­ minant |ast [ by a factor which is independent o f [asi]. Similarly, by consideration o f the columns, it is a multiple o f the determinant |bst | by a factor which is independent o f [bst], Hence it is o f the form k | ||b8t |, where k is independent o f both matrices. Comparing coefficients o f a11a22 a33b11b22b3Z it is clear that k = 1, and 1^116^1 = 1 ^ 6 « ] . The proof is clearly quite general and holds for square matrices o f any order.

5. , 2, 3' A = 3 ,-2 , 1 _ 4, 2, 1 2. A = “ 1, 3, 2” 2, 0, 5 , B = _ 6 , 1, 7 _ satisfies the equation A 3 — 23A — 40/ = 0. 6. Show that the matrices 1” “ 1 ,-2 , 1~ - 1, 3 ,-2 “ 1. £ = - 1, 2 ,-1 0, 0, = 1 , - 1. _ - 2, 4, - 2 _ 1 _ 1 ,-3 , 2 _1> ~ 1, satisfy A 2 = A, B2 = B, C2 — G, A B = BA — AC — CA = BC = CB = 0. 27 MATRICES Singular and Non-singular Matrices I f the columns o f a square matrix [ast] are linearly dependent then the matrix is said to be singular. From the theory o f determinants it follows that the condition for this is that the determinant o f the matrix is zero.

X A X , then the effect o f a transforma­ A b y TA T, where T is the matrix o f transformation. I f T is orthogonal, however, then T = T _1. Thus the matrix A is replaced by T~XA T. It is required, then, to transform the matrix A into diagonal form by means o f an orthogonal matrix. I f the quadratic form is tion is to replace the matrix QUADRATIC FORMS 49 The matrix A is a symmetric matrix, which is defined as a matrix which is equal to its transpose A = A. Symmetric matrices have the following fundamental property.

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A University Algebra: An Introduction to Classic and Modern Algebra by D. E. Littlewood


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