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Linear Algebra Matrices Question (1 Viewer)
- Thread starter phamtom44
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VBN2470
Well-Known Member
Q47 is a standard property of determinants and the proof is a bit long, if not tedious.
For Q48, if
is an 
orthogonal matrix, then 
. Take the determinant of both sides and use the result of Q47 to show that 
.
Hope it helps
For Q48, if
Hope it helps
seanieg89
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47. Largely depends on what you have learned / what definition of determinant you are working from.
Since you are just talking about square row echelon matrices (which are necessarily upper triangular), you can probably just use the fact that the determinant of an upper triangular square matrix is just the product of diagonal entries. (You could easily prove this fact inductively or otherwise if you like.)
Also if you multiply out two upper triangular square matrices, a simple computation shows that the resulting matrix is also upper triangular, and the diagonal entries are the product of the corresponding diagonal entries of the factor matrices.
This fact makes the result trivial.
48. Determinants are preserved under taking transpose, so=\det(QQ^T)=\det(Q)\det(Q^T)=\det(Q)^2.)
Since you are just talking about square row echelon matrices (which are necessarily upper triangular), you can probably just use the fact that the determinant of an upper triangular square matrix is just the product of diagonal entries. (You could easily prove this fact inductively or otherwise if you like.)
Also if you multiply out two upper triangular square matrices, a simple computation shows that the resulting matrix is also upper triangular, and the diagonal entries are the product of the corresponding diagonal entries of the factor matrices.
This fact makes the result trivial.
48. Determinants are preserved under taking transpose, so