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**Unformatted text preview: **Unit 16 Matrix Equations and the Inverse of a Matrix Now that we have defined matrix multiplication, we can better under- stand where the matrices we used in solving systems of linear equations came from. Consider a system of m equations in the n unknowns x 1 , x 2 , ..., x n . Let a ij be the coefficient of the j th variable in the i th equation, so that A = ( a ij ) is the coefficient matrix for this SLE. Let X be a column vector whose entries are the variables in the system, and let B be a column vector whose entries are the right hand side values of the equations. Then X is an n × 1 matrix and B is an m × 1 matrix. If we form the matrix product AX , we have: AX = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn · x 1 x 2 . . . x n = a 11 x 1 + a 12 x 2 + . . . + a 1 n x n a 21 x 1 + a 22 x 2 + . . . + a 2 n x n . . . . . . . . . . . . a m 1 x 1 + a m 2 x 2 + . . . + a mn x n and we see that AX is a column vector whose entries are the left hand sides of the equations. Thus, the matrix equation AX = B simply says that the i th entry of column vector AX must be equal to the i th entry of column vec- tor B , i.e., that for each equation in the system, the left hand side of the equation must equal the right hand side of the equation. 1 We see that AX = B is a matrix representation of the SLE. The aug- mented matrix ( A | B ), i.e., the coefficient matrix for the system with the column vector of right hand side values appended as an extra column, is just a form of short-hand for this matrix equation AX = B . For instance, the SLE: 3 x + 3 y + 12 z = 6 x + y + 4 z = 2 2 x + 5 y + 20 z = 10- x + 2 y + 8 z = 4 (from example 6 in Unit 14) can be written as the matrix equation: 3 3 12 1 1 4 2 5 20- 1 2 8 · x y z = 6 2 10 4 which says that: 3 x + 3 y + 12 z x + y + 4 z 2 x + 5 y + 20 z- x + 2 y + 8 z = 6 2 10 4 and the matrix equation can be written in short-hand as the augmented matrix: 3 3 12 6 1 1 4 2 2 5 20 10- 1 2 8 4 Notice that when we solve the SLE AX = B , the solution is a set of values for the unknowns in the column vector X . When we state a solution to a system of equations, it is usually more convenient to express it as a vector, rather than as a column vector. That is, if the solution to an SLE involving x , y and z is x = 1, y = 2 and z = 3, we generally write this as ( x, y, z ) = (1 , 2 , 3) rather than x y z = 1 2 3 2 Because an n × 1 column vector can also be written as an n-vector, we often give column vectors names in vector notation, rather than in matrix notation. That is, rather than talking about the column vectorsnotation....

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