Determinants

IIT‑JEE – Mathematics

Topic: Determinants

Goal of the notes – Give you a quick yet thorough reference that you can skim before the exam and that you can use to solve harder problems confidently.

1. Simple Explanation

A determinant is a scalar value that can be computed from a square matrix.
Intuitively it measures the “scaling factor” of the linear transformation represented by the matrix:

• Positive → orientation preserved,
• Negative → orientation reversed,
• Zero → transformation collapses the space (matrix is singular).

Determinants are central to many parts of IIT‑JEE maths: solving linear equations (Cramer’s rule), change of variables in integrals, eigenvalues, and many geometry/inequality problems.

2. Key Points

• Adding a multiple of one row/col to another → determinant unchanged.
• Multiplying a row/col by (k) → determinant multiplied by (k). | | Triangular Matrix | Determinant = product of diagonal entries. | | Block Matrices | If (A=\begin{bmatrix}B & 0\ 0 & C\end{bmatrix}) → (|A|=|B|\cdot|C|). | | Determinant of a Product | (|AB|=|A|\cdot|B|). | | Transpose | (|A^T|=|A|). | | Eigenvalues | (|A-\lambda I|=0) → characteristic polynomial. | | Vandermonde Matrix | (\displaystyle V(x_1,\dots,x_n)=\begin{vmatrix} 1&x_1&x_1^2&\cdots\ 1&x_2&x_2^2&\cdots\ \vdots&\vdots&&\ddots\end{vmatrix});
  (|V|=\prod_{1\le i<j\le n}(x_j-x_i)). | | Cramer’s Rule | For (AX=B): (x_i=\dfrac{|A_i|}{|A|}) where (A_i) is (A) with (i)-th column replaced by (B). | | Geometric Applications | Area of parallelogram = (|\det(v_1,v_2)|).
  Volume of parallelepiped = (|\det(v_1,v_2,v_3)|). | | Special Patterns |
• (a_{ij}=x_i^{j-1}) → Vandermonde.
• (a_{ij}=i+j) → can be expressed via row operations. | | JEE Tips |
• Use row/column operations to reduce to triangular form.
• Look for a row/column with many zeros or a common factor.
• For block or almost‑triangular matrices, break into smaller determinants.
• If the matrix depends on a variable (t), try to factor a common term from each row/col first. |

3. Worked Example

Problem (typical IIT‑JEE style)

[ A=\begin{bmatrix} 1 & 2 & 3 & 4\ 2 & 4 & 6 & 8\ 3 & 6 & 9 & 12\ 4 & 8 & 12 & 16 \end{bmatrix} ]

Find (|A|).

Step‑by‑Step Solution

1. Observe the pattern
   Every entry (a_{ij}=i\cdot j).
   Rows are multiples of each other:
   Row 2 = 2 × Row 1, Row 3 = 3 × Row 1, Row 4 = 4 × Row 1.

2. Apply Row Operations
   Subtract multiples of Row 1 from the others to create zeros:

   [ \begin{aligned} R_2 &\leftarrow R_2-2R_1 \ R_3 &\leftarrow R_3-3R_1 \ R_4 &\leftarrow R_4-4R_1 \end{aligned} ]

   Resulting matrix: [ \begin{bmatrix} 1 & 2 & 3 & 4\ 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0\ 0 & 0 & 0 & 0 \end{bmatrix} ]

3. Determinant of a Singular Matrix