Algebra and Multiple Linear Regression Part

In a previous chapter we noted that by augmenting the matrix of coefficients with unit matrix (i.e., one that has all the members equal to zero except on the main diagonal, where the members of the matrix equal unity), we could arrive at the solution to the simultaneous equations that were presented. Since simultaneous equations are, in one sense, a special case of regression (i.e., the case where there are no degrees of freedom for error), it is still appropriate to discuss a few odds and ends that were left dangling. 

We started in the previous chapter with the set of simultaneous equations  

expanding the matrix expression [A][B] into its full representation, we obtain 

From our previous chapter defining the elementary matrix operations, we recall the operation for multiplying two matrices the i, j element of the result matrix (where i and j represent the row and the column of an element in the matrix respectively) is the sum of cross-products of the /th row of the first matrix and the y th column of the second matrix (this is the reason that the order of multiplying matrices depends upon the order of appearance of the matrices - if the indicated ith row and y th column do not have the same number of elements, the matrices cannot be multiplied). 

Now let us apply this definition to the pair of matrices listed above. The first matrix ([A]) has three rows and three columns. The second matrix ( /i ) has three rows and one column. Since each row of [A] has three elements, and the single column of [B has three elements, matrix multiplication is possible. The resulting matrix will have three rows, each row resulting from one of the rows of matrix [A], and one column, corresponding to the single column in the matrix [ ]. 

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Matrix Algebra and Multiple Linear Regression Part 1... 

Matrix Algebra and Multiple Linear Regression Part 1 We can also rewrite the equation in matrix form as ... 

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