EXAMPLE 12 Without expanding the de...
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EXAMPLE 12 Without expanding the determinant, prove that a+b 2a + b 3a + b 2a + b 3a + b 4a + b = 0. 4a + b 5a + b 6a + b

11th - 12th Class
Maths
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soumon Let the given determinant be ( Delta ). Then, applying ( C_{3} rightarrow C_{3}-C_{2} ) we get: [ begin{aligned} Delta=&left|begin{array}{lll} a+b & 2 a+b & a 2 a+b & 3 a+b & a 4 a+b & 5 a+b & a end{array}right| =left|begin{array}{lll} a+b & a & a 2 a+b & a & a 4 a+b & a & a end{array}right| & text { [applying C }_{2} rightarrow C_{2}-C_{1} 1 end{aligned} ] [ =0 ] ( left[because C_{2} text { and } C_{3} ) are identical] right. Hence, ( Delta=0 )
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