Determinants & Properties
Program: SRMJEEE – Mathematics
1. Simple Explanation
A determinant is a scalar value that can be computed from a square matrix.
It tells us:
| Property | Meaning |
|---|---|
| Det = 0 | The rows (or columns) are linearly dependent → the matrix is singular (no inverse). |
| ** | det |
| Det(A·B) = Det(A)·Det(B) | Determinants multiply under matrix multiplication. |
Determinants are fundamental in:
- Solving linear systems (Cramer’s Rule)
- Finding inverses
- Calculating eigenvalues
- Geometric interpretations (area, volume)
2. Key Points & Properties
| # | Property | Formula / Condition | Notes |
|---|---|---|---|
| 1 | Definition | For an ( n \times n ) matrix ( A = [a_{ij}] ): (\displaystyle \det(A)=\sum_{\sigma\in S_n} \text{sgn}(\sigma)\prod_{i=1}^{n} a_{i,\sigma(i)}) |
Sum over all permutations; ( \text{sgn}(\sigma) ) is the parity of the permutation. |
| 2 | Row/Column Swapping | Swapping two rows (or columns) changes the sign: (\det(A_{swap}) = -\det(A)) |
Equivalent to multiplying by ( -1 ). |
| 3 | Row/Column Multiplication | Multiplying a row (or column) by scalar (k) scales the determinant by (k): (\det(kR_i) = k,\det(A)) |
|
| 4 | Row/Column Addition | Adding a multiple of one row to another does not change the determinant: (\det(A + kR_j \text{ to } R_i) = \det(A)) |
|
| 5 | Triangular Matrix | For an upper or lower triangular matrix, (\det(A)=\prod_{i=1}^{n} a_{ii}). | |
| 6 | Zero Row/Column | If any row or column is all zeros, (\det(A)=0). | |
| 7 | Determinant of Identity | (\det(I_n)=1). | |
| 8 | Multiplicative | (\det(AB)=\det(A)\det(B)). | |
| 9 | Transpose | (\det(A^T)=\det(A)). | |
| 10 | Adjugate | If (A) is invertible, (A^{-1} = \dfrac{1}{\det(A)},\text{adj}(A)). | |
| 11 | Cramer’s Rule | For (AX=B) with (A) invertible, the (i^{th}) variable: (x_i = \dfrac{\det(A_i)}{\det(A)}) where (A_i) replaces column (i) of (A) with (B). |
Works only if (\det(A)\neq0). |
| 12 | Linear Dependence | If two rows are linearly dependent, (\det(A)=0). | Equivalent to a row being a scalar multiple of another. |
| 13 | Expansion by Minors (Laplace) | (\displaystyle \det(A)=\sum_{j=1}^{n}(-1)^{i+j} a_{ij}\det(M_{ij})) for any fixed row (i) or column (j). | (M_{ij}) is the ((n-1)\times(n-1)) sub‑matrix after deleting row (i) and column (j). |
| 14 | Sarrus Rule (3×3) | For (A=\begin{bmatrix}a&b&c\ d&e&f\ g&h&i\end{bmatrix}): (\det(A)=a e i + b f g + c d h - c e g - b d i - a f h). |
Handy shortcut for 3×3. |
| 15 | Determinant as Volume | If (A) has column vectors (v_1,\dots,v_n), the absolute value ( | \det(A) |
3. Worked Example
Problem
Find the determinant of
[ A=\begin{bmatrix} 2 & 0 & 1\ -1 & 3 & 2\ 4 & -2 & 5 \end{bmatrix} ]
Solution
We'll use cofactor expansion along the first row (any row/column works; choose the one with zeros for less work).
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Identify minors and cofactors
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(a_{11}=2), minor (M_{11}) by deleting row 1, col 1:
[ M_{11}=\begin{bmatrix}3 & 2\ -2 & 5\end{bmatrix} ]
(\det(M_{11}) = 3\cdot5 - 2(-2) = 15 + 4 = 19).
Cofactor (C_{11}=(-1)^{1+1}\det(M_{11})=+19).
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(a_{12}=0) → its cofactor contributes 0.
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(a_{13}=1), minor (M_{13}) (delete row 1, col 3):
[ M_{13}=\begin{bmatrix}-1 &
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