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Question:

Using properties of determinants, prove that ⎡⎢⎣x+yxx5x+4y4x2x10x+8y8x3x⎤⎥⎦=x3

Solution:

To prove, ∣∣∣∣x+yxx5x+4y4x2x10x+8y8x3x∣∣∣∣=x3
LHS=∣∣∣∣xxx5x4x2x10x8x3x∣∣∣∣+∣∣∣∣yxx4y4x2x8y8x3x∣∣∣∣=x3∣∣∣∣1115421083∣∣∣∣+yx2∣∣∣∣111442883∣∣∣∣
Applying C1→C1−C2,C2→C2−C3 in the first determinant=x3∣∣∣∣001122253∣∣∣∣+yx2×0
As the first two columns of the 2nd determinant are same.
Expanding the first determinant through R1=x3.1.∣∣∣1225∣∣∣=x3(5𕒸)=x3= RHS
Hence proved.