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I S. Y

A ,

Q . THE PROPERTIES OF DETERMINANTS

a. An n×nmatrix Ais called orthogonal if AAt=I. If Ais orthogonal show that det(A) = ±1.

A

By the properties of determinant, det(AAt) = det(A)det(At) = det(A)det(A) = det(A)2=det(I) = 1. So,

det(A) = ±1.

b. An n×nmatrix A is called skew-symmetric if At=−A. Show that if Ais skew-symmetric and nis

an odd positive integer, then Ais not invertible.

A

By the properties of determinant, det(At) = det(−A)Ñdet(A) = det(−A)Ñdet(A) = (−1)ndet(A)Ñ

det(A) = −det(A). So, we get det(A) = 0 which implies that Ais not invertible. Note that −Ameans that

EVERY ROW of Ais multiplied by -1.

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Q . CRAMER’S RULE, VOLUME, AND LINEAR TRANSFORMATIONS

Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices are (1,0,−2),

(1,2,4), and (7,1,0).

A

We need calculate the determinant

117

021

−240

,

then take the absolute value to get the volume. So, the volume of the parallelepiped is |22|= 22.

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