qr code vb.net library Problem 234 Find the inverse of the matrix 2 6 4 0 6 8 0 1 2 4 3 7 35 3 in .NET framework

Print Code 128 Code Set C in .NET framework Problem 234 Find the inverse of the matrix 2 6 4 0 6 8 0 1 2 4 3 7 35 3

Problem 234 Find the inverse of the matrix 2 6 4 0 6 8 0 1 2 4 3 7 35 3
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CHAPTER 11 Matrix Algebra. Networks
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Problem 235 Suppose one set of variables, x; y; z, is related to another set, r; s; t, by the three simultaneous linear equations 3x 4y z r 2x y 5z s 4x 6y 2z t It follows that it is also possible to express set 1 in the equivalent form ar bs ct x dr es ft y gr hs it z provided, of course, that the constant coe cients, a through i, are given the correct values. Problem: rst express set 1 in matrix form in the manner of eq. (476), and then, by use of the inverse operation, nd the values of a through i that will permit the second set of equations to be written in place of the rst set. set 2 set 1
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Some Properties of the Unit Matrix
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The unit or identity matrix, denoted by I, is any square matrix in which all elements of the main diagonal are equal to 1, all other elements being equal to zero. (See Fig. 272 as an example.) Let us now state that the PRODUCT of any square matrix A, of order n, and a unit matrix of the same order n, is equal to the matrix A; that is, AI A. The truth of this statement will be clear from a study of the following example, in which a square thirdorder matrix A multiplies a third-order unit matrix. Thus, using the procedure of matrix multiplication as de ned in section 11.2, you should verify that the following multiplication is correct. 2 32 3 2 3 1 0 0 a11 a12 a13 a11 a12 a13 6 76 7 6 7 4 a21 a22 a23 54 0 1 0 5 4 a21 a22 a23 5 0 0 1 a31 a32 a33 a31 a32 a33 that is, AI A From the above it s clear that if A is ANY square matrix of order n, and if I is a unit matrix of the same order n, then AI A. Now reverse the order of multiplication in the above example; doing this, and again keeping the de nition of matrix multiplication in mind, you will nd it is also true that IA A. Thus, if A is a square matrix of order n and I is a unit matrix of the same order n, then A and I are commutative in multiplication, and we have the useful relationship AI IA A 486 Thus the unit matrix behaves, in matrix multiplication, much like the number 1 does in ordinary algebraic multiplication. Another useful relationship can be found as follows. Let us begin with the matrix equation AX Y
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Multiply both sides by A 1 :
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Matrix Algebra. Networks
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487
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A 1 AX A 1 Y From eq. (480) we know that X A 1 Y From eq. (486), X IX, and hence IX A 1 Y
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488
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Comparison of eqs. (487) and (488) shows that A 1 A I; likewise, AA 1 I, and we thus have the useful fact that the product of a matrix A and its inverse equals a unit matrix I; that is, AA 1 A 1 A I 489
Problem 236 If we have found the inverse of a given matrix A, then eq. (489) can be used to check the correctness of our work. This is true because, if our work is correct, it has to be true that AA 1 I, as eq. (489) states. In the example in section 11.3 we found that if 2 3 2 1 4 6 7 A 4 1 5 35 2 0 then A 1 5 1 1 6 4 7 6 37 10 2 2 1 3 23 7 10 5 9
Your problem here is to double-check the above result by verifying, by actual multiplication, that eq. (489) is satis ed.
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