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Exercises 4, , Ai js a non-singular matrix of otder 4, Determine the rank of the matrix, AT, LRA’, Giadj A, G0 2A., , yfthe rank of a real symmetric matrix be 1 show that the diagouel elements, of the matrix cannot be all zero., , ‘M, , p Prove that the rank of a real skew-symmetric matrix cannot be I., , pKis an n X n matrix of rank n — 1. Prove that the rank of adj A is 1., [Hint. Since det A = 0, rank(adj A) <1, by Theorem 2.4.3. Since rank of A is n— 1,, there is at least one non-zero minor of A of order n—1 and therefore rank(adj A) > 1.], , Find all real x for which the rank of the matrix is less than 4., , zxi1i1ii 1+2 2 3 4, A Liagsndiol pes 1 2+2 3 4 4, 11<¢e24it1 , 1 2 342 4 ,, 111 <¢2 1 2 3 4+2z, l+a 2 3 4, woe] 2 l+a 3 4, Hey 2 3 1+2z 4, 2 3 4 1l+2, Jind all real \ for which the rank of the matrix A is 2., 1231 i "1 1 1, Min(a 59d ) wfa-(1 22 A]., 11.6 At+i1 5 7 1 xX, Kt 010 ; ., ‘“tA={ 0 0 1. | find the rank of the matrix :, 100, , FTAS A, Gar + AP + AP, , b

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ti, 104 HIGHER ALGEBRA, , Ww 7 Reduce the matrix A to the fully reduced normal form and. find non-singula:, matrices P and Q such that PAQ is the fully reduced normal form., , 2 1 2: 102 8, graa=(101), pya=(214 6). >, —_ Li Le 307 9, , 1, , /Use elementary row operations on A to obtain A7! where A is20 0\.° 2 0 0 12 3, , {0 30), i 4 3 0 |, SO oe |., 00 4 64 1 001, , ~ Express A’as the product of elementary matrices., , poo state oe), , 0 0 2, , “11. A and B are equivalent square matrices. Prove that, , firihe matrices A* and B* are equivalent;, Si the matrices A? and B? are not necessarily equivalent;, , iid’ the matrices AB and BA are equivalent if one of A and B be nonsingular;, , (iv) the matrices AB and BA are not necessarily equivalent if none of A, and B be non-singular.