By D. A. Suprunenko; translated by K.A. Hirsch
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8 with all real entries. Show that the solutions to det(A − λI) = 0 are real A. If and only if (trA)2 ≥4 det A B. If A is symmetric C. 3, show how to obtain x2 in the formula for Cramer’s rule by the elimination process mentioned there. 1 Introduction Two methods of finding the inverse of a matrix will be taken up in this section. The first uses the adjoint of a matrix and gives a formula in terms of it. The second is based on elementary row operations. We will then present a method of solving a system of equations that uses a matrix inverse.
Generalized inverses will be studied in Part III. 14 Show that for any matrix X, X′X, and XX′ are symmetric matrices. 15 Let A and B be two 2 × 2 matrices where the rows and the columns add up to 1. Show that AB has this property. 16 Consider the linear model y11 = µ + α1 + β1 + ε11 y12 = µ + α1 + β2 + ε12 y21 = µ + α 2 + β1 + ε 21 y22 = µ + α 2 + β2 + ε22 . A. Tell what the matrices should be for a model in the form Y = Xγ + ε where Y is 4 × 1, X is 4 × 5, γ is 5 × 1, and ε is 4 × 1. B. Find X′X.
The elements of the matrix Thus, n det(A ) i = j adj(A)A are ∑ a ik C kj = . 3) i≠ j k =1 0 adj(A) A = det(A)I. 4) Let Aj represent the n × n matrix with the elements of A everywhere but the jth column. The jth column consists of the n-dimensional column vector b. The determinant of Aj is the scalar product of the jth row of adj(A) and b. 4 Solution of a System of Two Equations in Two Unknowns by Cramer’s Rule Consider the system of equations 2 x + 3y = 12 3x – 4 y = 1. SOLUTION TO LINEAR EQUATIONS USING DETERMINANTS 21 12 Observe that for x the vector is in the first column of the determinant in the 1 numerator and for y the same vector is in the second column of the numerator.
Soluble and nilpotent linear groups. by D. A. Suprunenko; translated by K.A. Hirsch