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Chapter 6: Q51E (page 292)
Find the determinant of the\((2n) \times (2n)\)matrix
\(A = \left( {\begin{array}{*{20}{c}}0&{{I_n}}\\{{I_n}}&0\end{array}} \right).\)
Therefore, the determinant of the given matrix is given by,\(\det A = {( - 1)^n}\).
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A square matrix is called a permutation matrix if each row and each column contains exactly one entry 1, with all other entries being 0 . Examples are , and the matrices considered in Exercises 53 and 56 . What are the possible values of the determinant of a permutation matrix?
In Exercises 62 through 64, consider a function D from to that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume that.
62. Show thatfor anymatrix whose two columns are equal.
The matrix is invertible for all positive constantsk.
If Ais any n x n matrix, then.
IfA is a matrix whose entries are all 1 or -1 , then must be divisible by 8 (i.e., for some integer k).
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