Number of nonzero entries most 0s examples along

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number of nonzero entries (most 0s) Examples: along column 3 12 3 4 230 1 3 4 0 1 0 103 = 3 along row 3 2 3 1 341 0 1 3 = 3 21 3 1 + 3 2 3 3 4 ! = 3 ( ( 1 )+ 3 ( 1 )) = 6 along row 1 0100 0001 0010 1000 = along row 1 00 1 0 1 0 100 = 0 1 1 0 = ( 1 )= 1
If A has a row or column of 0 s then det ( A )= 0 (expand along that row or column!) If A is (upper or lower) triangular then det ( A a 11 a 22 ··· a nn (product of diagonal coefficients) Example: along row 1 4000 5 200 63 4 0 70 1 3 = 4 along row 1 3 4 0 0 1 3 = 4 · 2 4 0 1 3 = 4 · 2 · 4 · 3 = 96
Properties of determinants School of Mathematical & Statistical Sciences
Determinants via row/column operations Row operations afect determinants as Follows: A row swap changes the sign oF a determinant A r i r k −−−−−→ B = A A row scaling multiplies a determinant by the value oF the scaling A r k α r k −−−−−−−→ B = α A Subtracting a multiple oF a row From another one leaves a determinant unchanged A r i r i α r k −−−−−−−−−−→ B = A Column operations have a similar efect
Examples: v v v v v v 1 2 3 4 5 6 7 8 9 v v v v v v r 2 r 2 - 4 r 1 ========== r 3 r 3 - 7 r 1 v v v v v v 1 2 3 0 3 6 0 6 12 v v v v v v r 3 r 3 - 2 r 2 v v v v v v 1 2 3 0 3 6 0 0 0 v v v v v v = 0 v v v v v v v v 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 v v v v v v v v r 1 r 4 ====== v v v v v v v v 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 v v v v v v v v r 2 r 4 v v v v v v v v 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 v v v v v v v v = det( I 4 ) = 1 v v v v v v v v 1 2 3 4 2 3 0 1 3 4 0 1 0 1 0 3 v v v v v v v v c 1 c 3 ======= v v v v v v v v 3 2 1 4 0 3 2 1 0 4 3 1 0 1 0 3 v v v v v v v v c 2 c 3 v v v v v v v v 3 1 2 4 0 2 3 1 0 3 4 1 0 0 1 3 v v v v v v v v r 3 r 3 - 3 2 r 2 v v v v v v v v 3 1 2 4 0 2 3 1 0 0 1 2 1 2 0 0 1 3 v v v v v v v v r 4 r 4 + 2 r 3 v v v v v v v v 3 1 2 4 0 2 3 1 0 0 1 2 1 2 0 0 0 2 v v v v v v v v = 3 · 2 · ( 1 2 ) · 2 = 6
If A has identical rows or columns then det( A ) = 0 1 2 3 1 2 3 9 8 7 r 1 r 1 - r 2 ======= 0 0 0 1 2 3 9 8 7 = 0 If a row/column of A is a linear combination of other rows/columns then det( A ) = 0 1 2 3 4 5 6 7 8 9 c 2 c 2 - 1 2 c 1 - 1 2 c 3 ============ 1 0 3 4 0 6 7 0 9 = 0 The column operation c 2 c 2 - 1 2 c 1 - 1 2 c 3 is equivalent to the 2 independent operations c 2 c 2 - 1 2 c 1 and c 2 c 2 - 1 2 c 3
Example with parameter: For what real value(s) of t does v v v v v v 1 1 1 1 t t 2 1 t 2 t 4 v v v v v v = 0? v v v v v v 1 1 1 1 t t 2 1 t 2 t 4 v v v v v v r 2 r 2 - r 1 ======= r 3 r 3 - r 1 v v v v v v 1 1 1 0 t 1 t 2 1 0 t 2 1 t 4 1 v v v v v v factor ( t - 1) out of r 2 =============== factor ( t 2 - 1) out of r 3 ( t 1)( t 2 1) v v v v v v 1 1 1 0 1 t + 1 0 1 t 2 + 1 v v v v v v r 3 r 3 - r 2 ======= ( t 1)( t 2 1) v v v v v v 1 1 1 0 1 t + 1 0 0 t 2 t v v v v v v = ( t 1)( t 2 1)( t 2 t ) = t ( t + 1)( t 1) 3 = 0 for t = 0 , 1 , 1 Check For t = 0 columns 2 and 3 are identical 1 columns 1 and 3 are identical 1 all columns are identical c
Additional example: Let A = r 1 r 2 r 3 with det( A ) = 6. Find det r 1 + r 2 r 2 + 3 r 1 r 3 v v v v v v r 1 + r 2 r 2 + 3 r 1 r 3 v v v v v v row 2 row 2 - 3row 1 ================== (aim: eliminate r 1 from row2) v v v v v v r 1 + r 2 2 r 2 r 3 v v v v v v Factor

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