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Unformatted text preview: Then we can row reduce the A part and D part of our block matrix independently so its determinant is det ( A ) det ( D ). (b) Consider 1 0 0 1 r 0 1 1 0 r 1 0 0 1 . Here B and C both have determinant 0 while A and D have determinant r , so  A  D    C  B  = r 2 . The ﬁrst and last rows of 1 0 0 1 r 0 1 1 0 r 1 0 0 1 are the same, so row reduction gives a row of zeros and the matrix has determinant 0. As long as r 6 = 0 the values are diﬀerent. (c) In the example above, ADCB = ± 1 0 r ²± r 0 1 ²± 1 0 1 0 ²± 0 1 0 1 ² = ± r r ²± 0 1 0 1 ² = ± r1 r1 ² which has determinant r ( r1) and is nonzero as long as r 6 = 0 , 1. But our big 4 × 4 determinant is 0....
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 '05
 HUI
 Math, Linear Algebra, Algebra, Zero, Row, ﬁrst row

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