6. Let Z be the m X p matrix (Zai)' where p ;5; m and the Zai are independently distributed as N(O.I). let S = Z/Z. and let SI be the matrix obtained by omitting the last row and column of S. Then the ratio of determinants ISVISd has a x 2-distribution with m - p + 1 degrees of freedom. [Let q be an orthogonal matrix (dependent on Zl1 .... . Zml) such that (Zl1 .. . Zml)Q/ = (R 0 .. . 0). where R2 = L::=IZ~I Then R 0 .. . Z~2 ) ( Zi2 ... Zip Zi2 Zi2 ... Zi2 Zip S = Z/Q/QZ = I Zip Zip ... Z:'pJ \ 0 Z:'2 ... Z:'p where the Z;; denote the transforms under Q. The first of the matrices on the right-hand side is equal to the product ($)(*)' where Z* is the (m - 1) X (p - 1) matrix with elements Z;; (a = 2•. . . • m; i = 2•. . . • p). I is the (p -1) X (p -1) identity matrix. Zr is the column vector (ZI"2' " Zl"p)'. and 0 indicates a row or column of zeros. It follows that lSI is equal to R2 multiplied by the determinant of Z*/Z*. Since SI is the product of the m X (p - 1) matrix obtained by omitting the last column of Z multiplied on the left by the transpose of this m X (p - 1) matrix. lSI!is equal to R2 multiplied by the determinant of the matrix obtained by omitting the last row and column of Z* /Z* . The ratio ISl/lSd has therefore been reduced to the corresponding ratio in terms of the Z;; with m and p replaced by m - 1 and p - 1. and by induction the problem is seen to be unchanged if m and p are replaced by m - k and p - k for any k < p . In particular. ISl/lSd can be evaluated under the assumption that m and p have been replaced by m - (p - 1) and p - (p - 1) = 1. In this case. the matrix Z / is a row matrix (Zl1 ' " Zm -pHI); the determinant of S is lSI = L~I:f+IZ~I' which has a X~I p +I-distribution; and since S is a 1 X 1 matrix. lSI1is replaced by 1.]