JEE Mathematics Determinants Notes Set 05

Determinant and Its Value

A determinant is a number assigned to a square array. \(\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc\). For a third order determinant, expand along any row or column using minors and cofactors with the sign pattern \(+\,-\,+\), \(-\,+\,-\), \(+\,-\,+\): \(\Delta=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\). The sum of products of the entries of one row with the cofactors of another row is zero.

Properties

1. The value is unchanged when rows and columns are interchanged. 2. Interchanging two rows (or columns) changes the sign. 3. If two rows (or columns) are identical or proportional, the value is 0. 4. Multiplying one row by k multiplies the determinant by k, so \(|kA|=k^n|A|\) for an order n matrix. 5. If each entry of a row is a sum of two terms, the determinant splits into the sum of two determinants. 6. Adding a multiple of one row to another leaves the value unchanged. 7. \(|AB|=|A||B|\).

Useful Techniques

Row-sum trick: if every row has the same sum S, add all columns to the first column and take out S. Alternating determinants such as \(\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix}=(b-a)(c-a)(c-b)\) vanish when two variables are equal, so the factors \((a-b)\), \((b-c)\), \((c-a)\) can be read off. A skew-symmetric determinant of odd order is 0. Also \(\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}=-(a^3+b^3+c^3-3abc)\).

Worked Examples

Example 1. \(\begin{vmatrix}y+k&y&y\\y&y+k&y\\y&y&y+k\end{vmatrix}\): add all rows to the first row to get \((3y+k)\) in each entry; subtracting leaves \(k^2(3y+k)\).

Example 2. The roots of \(\begin{vmatrix}1&4&20\\1&-2&5\\1&2x&5x^2\end{vmatrix}=0\): subtract row 2 from rows 1 and 3 to get \(-30(x-2)(x+1)\), so \(x=2\) or \(-1\).

Example 3. For a 3 by 3 matrix with \(|A|=4\), \(|3A|=27\times4=108\).

Quick Tips

Look for equal row sums, repeated or proportional lines and common factors before expanding. Test an identity with simple numbers to choose between options.