Question
Evaluate the determinant .
Solution — Step by Step
Subtract from and from (this doesn’t change the determinant value when we replace a row with itself plus a multiple of another row):
gives . gives .
The matrix is now:
Rows 2 and 3 are proportional (). When two rows are proportional, the determinant is zero.
Final answer: the determinant is .
Why This Works
A determinant is zero when the rows (or columns) are linearly dependent. After our row operations, two rows became proportional, signalling dependence. The original matrix’s rows form an arithmetic progression along each column, which guarantees linear dependence — Row 2 = (Row 1 + Row 3)/2.
This pattern (three rows in AP) always gives determinant zero. Recognising this saves a lot of expansion work.
Alternative Method
Expand along the first row:
Same answer, more arithmetic.
If rows or columns are in arithmetic progression, the determinant is zero. Check for AP patterns first — they appear in JEE Main 1-2 times a year.
Common Mistake
Computing a determinant straight from cofactor expansion when row operations would simplify dramatically. The exam clock matters — recognise patterns first, expand only as a last resort. Also, some students do row operations incorrectly: only preserves the determinant. Multiplying a row by multiplies the determinant by .