Matrices and Determinants 2 Question 42

46. Without expanding a determinant at any stage, show that

|x2+xx+1x22x2+3x13x3x3=xA+Bx2+2x+32x12x1| =xA+B

where A and B are determinants of order 3 not involving x.

(1982,5M)

Show Answer

Solution:

  1. Let Δ=|x2+xx+1x22x2+3x13x3x3x2+2x+32x12x1|

Applying R2R2(R1+R3), we get

Δ=|x2+xx+1x2400x2+2x+32x12x1|

Applying R1R1+x24R2

and R3R3+x24R2, we get

Δ=|xx+1x24002x+32x12x1|

Applying R3R32R1=|x+0x+1x2400333|

|xxx400333| + |012400333|

= x |111400333| + |012400333|

Δ=Ax+B

where, A=|111400333|

and B=|012400333|



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