8 det A det A t Tutorials 536 1 Find the value of x such that a x 3 5 3 x 2 45

8 det a det a t tutorials 536 1 find the value of x

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8. det( A ) = det( A t ) Tutorials 5.3.6. 1. Find the value of x such that: (a) x - 3 5 - 3 x - 2 = 45 (b) x - 1 2 x 1 - 1 0 2 - 1 - x = 12 127
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2. Use properties of determinants to evaluate: (a) 3 - 4 - 2 6 - 4 - 1 9 2 3 (b) 1 k k 2 1 k k 2 1 k k 2 (c) 3 3 0 5 2 2 0 - 2 4 1 - 3 0 2 10 3 2 (d) 4 0 0 1 0 3 3 3 - 1 0 1 2 4 2 3 2 2 4 2 3 - 3 - 1 2 0 9 3. Show that the value of the following determinant is independent of x . sin x cos x 0 - cos x sin x 0 sin x - cos x sin x + cos x 1 = 0 . 5.4 Matrix Inversion Definition 5.4.1. An n × n square matrix A is invertible (non-singular) if there exist a matrix D such that AD = DA = I n . We say D is an inverse matrix of A and D = A - 1 . Note 5.4.2. The following facts are important: If A - 1 exists, then it is unique and square of the same order as A . 128
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An invertible matrix and its inverse commute with respect to matrix mul- tiplication i.e. AA - 1 = A - 1 A = I n . A non-square matrix can never be invertible, and also among square ma- trices there are infinitely many that are not invertible. There is a symmetric relationship between a matrix and it’s inverse; i.e. if B is the inverse of A , then A is also an inverse of B . If A - 1 exists, then the system of linear equations with A as a coefficient matrix ALWAYS has a unique solution, whatever the right hand side. We can actually write the equation as AX = B where X is the column matrix of variables x, y, z, etc and B is the column matrix of the right hand side. Then AX = B implies A - 1 AX = A - 1 B , then I n X = A - 1 B , hence X = A - 1 B . This will yield the unique solution. (We’ll touch this in detail later). We will study two methods of finding the inverse of a matrix. The first method involves Gauss-Jordan operation and the second is the adjoint method . A. Gauss-Jordan elimination method: We will make use of the following algorithm called Matrix inversion algo- rithm in order to evaluate the inverse using the Gauss-Jordan elimination. NB: Matrix Inversion Algorithm: If A is a (square) invertible matrix, there exists a sequence of elementary row operations that carry A to the identity matrix I of the same size, written A I . This same series of row operations carries I to A - 1 ; that is, I A - 1 . The algorithm can be summarized as follows: [ A | I ] [ I | A - 1 ] where the row operations on A and I are carried out simultaneously. Example 5.4.3. Use the Gauss-Jordan elimination to find the inverse of 129
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the following matrices: ( a ) A = - 2 3 4 - 5 ; ( b ) B = 2 7 1 1 4 - 1 1 3 0 (a) ( A | I ) = - 2 3 4 - 5 1 0 0 1 = - 2 3 0 1 1 0 2 1 R 2 + 2 R 1 = - 2 0 0 1 - 5 - 3 2 1 R 1 - 3 R 2 = 1 0 0 1 5 2 3 2 2 1 ( - 1 2 ) × R 1 A - 1 = 5 2 3 2 2 1 . (b) ( A | I ) = 2 7 1 1 4 - 1 1 3 0 1 0 0 0 1 0 0 0 1 130
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First interchange row 1 and 2. 1 4 - 1 2 7 1 1 3 0 0 1 0 1 0 0 0 0 1 R 2 R 1 1 4 - 1 0 - 1 3 0 - 1 1 0 1 0 1 - 2 0 1 - 1 1 R 2 - 2 R 1 R 3 - R 1 1 4 - 1 0 1 - 3 0 - 1 1 0 1 0 - 1 2 0 0 - 1 1 - R 2 1 0 11 0 1 - 3 0 0 - 2 4 - 7 0 - 1 2 0 - 1 1 1 R 1 - 4 R 2 R 3 + R 2 1 0 11 0 1 - 3 0 0 1 4 - 7 0 - 1 2 0 1 2 - 1 2 - 1 2 - 1 2 R 3 1 0 11 0 1 - 3 0 0 1 - 3 2 - 3 2 11 2 1 2 1 2 - 3 2 1 2 - 1 2 - 1 2 R 1 - 11 R 3 R 2 + 3 R 3 Hence A - 1 = - 3 2 - 3 2 11 2 1 2 1 2 - 3 2 1 2 - 1 2 - 1 2 = 1 2 - 3 - 3 11 1 1 - 3 1 - 1 - 1 B. The Adjoint Method: Definition 5.4.4.
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