I have added theorem 3.8 as per the question In excereße 6 Find the Pniverse of thegiven matriu (pfft exists) using Theorem 3.86./1/√2.-1/√21N21/√2 166Theorem 3.8Chapter 3 MatricesIf A =andb]then A is invertible if ad-bc0, in which caseA¹ =If ad bc= 0, then A is not invertible.Similarly,€ [a b]₁The expression ad-bc is called the determinant of A, denoted det A. The formulafor the inverse of(when it exists) is thustimes the matrix obtained bydet Ainterchanging the entries on the main diagonal and changing the signs on the othertwo entries. In addition to giving this formula, Theorem 3.8 says that a 2 x 2 matrixA is invertible if and only if det A # 0. We will see in Chapter 4 that the determinantcan be defined for all square matrices and that this result remains true, although thereis no simple formula for the inverse of larger square matrices.Proof Suppose that det A = ad-bc # Then[a b][-d-d]=ad-bc-c[ad-bc ab + ba] [ad-bc=cd-de -cb + da0[-][a] =det A[i]Since det A # 0, we can multiply both sides of each equation by 1/det A to obtain[1-6]-01-01[Note that we have used property (d) of Theorem 3.3.] Thus, the matrix1 ddet A-cd1det A-csatisfies the definition of an inverse, so A is invertible. Since the inverse of A is unique,by Theorem 3.6, we must have1A = ded -bIn the first case,A = [a b]-[aca[ac/aConversely, assume that adbc = 0. We will consider separately the cases wherea 0 and where a = 0. If a # 0, then d= bc/a, so the matrix can be written asaw-ckawd-bbbc/a]where k = c/a. In other words, the second row of A is a multiple of the first. Referringto Example 3.23(b), we see that if A has an inversethenka kband the corresponding system of linear equations+ byky-61+ kbykax0ad - be] = det A[!]= 1+bz 0= 0+ kbz 1has no solution. (Why?)If a = 0, then ad bc = 0 implies that bc = 0, and therefore either b or c is 0.Thus, A is of the formо[2] or [1]d][oo]• [[ ]]-[9]+[1]have an inverse. (Verify this.)Consequently, if ad-bc= 0, then A is not invertible.Similarly,2cannot

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In excercise 6 Find the inverse of the given matriu (pfft exists) using Theorem 3.8 6:11/√2. -1/2 -1/√2 1/√2