Matrices and Determinants Formula Sheet
All key Matrices and Determinants formulas for JEE Main & Advanced: matrix operations, transpose, determinants, minors, cofactors, adjoint, inverse, Cramer's rule.
Every must-know result from the Matrices and Determinants chapter, grouped for fast last-minute revision. Everything here is pulled from the chapter topics — use it as your one-page reference before the exam.
Matrix Basics
A matrix is a rectangular array of numbers. Element $a_{ij}$ sits in row $i$, column $j$ (remember: Row then Column).
$$\boxed{\text{Order} = m \times n \quad (m \text{ rows} \times n \text{ columns})}$$Compact notation: $A = [a_{ij}]_{m \times n}$
Trace (sum of diagonal elements of a square matrix):
$$\boxed{\text{tr}(A) = \sum_{i=1}^{n} a_{ii}}$$Equality of matrices:
$$\boxed{A = B \iff [a_{ij}]_{m \times n} = [b_{ij}]_{m \times n}}$$(Same order and all corresponding elements equal.)
Memory hook: $a_{ij}$ = “RC Cola” — Row first, Column second.
Types of Matrices
| Type | Defining condition |
|---|---|
| Row matrix | Order $1 \times n$ |
| Column matrix | Order $m \times 1$ |
| Square matrix | Rows = columns ($n \times n$) |
| Diagonal | $a_{ij} = 0$ when $i \neq j$ |
| Scalar | Diagonal matrix with all diagonal elements equal |
| Identity $I_n$ | Diagonal $= 1$, rest $= 0$; $AI = IA = A$ |
| Null / zero $O$ | All elements $= 0$; $A + O = A$ |
| Upper triangular | $a_{ij} = 0$ when $i > j$ |
| Lower triangular | $a_{ij} = 0$ when $i < j$ |
| Symmetric | $A = A^T$, i.e. $a_{ij} = a_{ji}$ |
| Skew-symmetric | $A = -A^T$, i.e. $a_{ij} = -a_{ji}$ (diagonal must be $0$) |
Matrix Algebra
Addition, Subtraction, Scalar Multiplication
Defined only for matrices of the same order:
$$\boxed{(A \pm B)_{ij} = a_{ij} \pm b_{ij}} \qquad \boxed{(kA)_{ij} = k\, a_{ij}}$$Addition is commutative, associative, has identity $O$ and inverse $-A$.
Scalar multiplication properties: $k(lA) = (kl)A$, $\;k(A+B) = kA + kB$, $\;(k+l)A = kA + lA$, $\;1\cdot A = A$, $\;(-1)A = -A$.
Matrix Multiplication
Possible only if columns of $A$ = rows of $B$. If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$.
$$\boxed{(AB)_{ik} = \sum_{j=1}^{n} a_{ij}\, b_{jk}}$$| Property | Result |
|---|---|
| Commutative? | $AB \neq BA$ in general (NOT commutative) |
| Associative | $(AB)C = A(BC)$ |
| Distributive (left) | $A(B+C) = AB + AC$ |
| Distributive (right) | $(A+B)C = AC + BC$ |
| Identity | $AI = IA = A$ |
| Zero | $AO = OA = O$ |
| Scalar | $k(AB) = (kA)B = A(kB)$ |
- $AB = O$ does not imply $A = O$ or $B = O$ (no zero-product property).
- $AB = AC$ does not imply $B = C$ unless $A$ is invertible ($|A| \neq 0$).
Powers of a Matrix
For a square matrix: $A^n = \underbrace{A \cdot A \cdots A}_{n \text{ times}}$
| Result | Condition |
|---|---|
| $A^m A^n = A^{m+n}$ | always |
| $(A^m)^n = A^{mn}$ | always |
| $A^m B^m = (AB)^m$ | only if $AB = BA$ |
| $I^n = I$ | always |
| $D^n = \text{diag}(a^n, b^n, \dots)$ | diagonal $D$ |
Nilpotent: $A^k = O$ for some $k$. Idempotent: $A^2 = A$. Involutory: $A^2 = I$.
Transpose
$$\boxed{A^T = [a_{ji}]_{n \times m}}$$| Property | Result |
|---|---|
| Double transpose | $(A^T)^T = A$ |
| Addition | $(A+B)^T = A^T + B^T$ |
| Scalar | $(kA)^T = kA^T$ |
| Product (order reverses) | $(AB)^T = B^T A^T$ |
| Extended | $(ABC)^T = C^T B^T A^T$ |
Orthogonal matrix: $A^T A = AA^T = I$ (so $A^{-1} = A^T$).
Symmetric / Skew-Symmetric Decomposition
Every square matrix splits uniquely into a symmetric and a skew-symmetric part:
$$\boxed{A = \underbrace{\frac{A + A^T}{2}}_{\text{symmetric}} + \underbrace{\frac{A - A^T}{2}}_{\text{skew-symmetric}}}$$Determinants: Evaluation
Determinant is a scalar defined only for square matrices; denoted $|A|$ or $\det(A)$.
2×2:
$$\boxed{\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc}$$3×3 (expansion along Row 1):
$$\boxed{\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)}$$3×3 (Sarrus rule): sum of three “down-right” diagonal products minus three “down-left” anti-diagonal products.
$$|A| = (a_1 b_2 c_3 + b_1 c_2 a_3 + c_1 a_2 b_3) - (c_1 b_2 a_3 + a_1 c_2 b_3 + b_1 a_2 c_3)$$Special / Instant Determinants
| Matrix | Determinant |
|---|---|
| Diagonal | Product of diagonal elements |
| Triangular (upper/lower) | Product of diagonal elements |
| Identity | $\|I_n\| = 1$ |
| Zero matrix | $\|O\| = 0$ |
| Any zero row or column | $\|A\| = 0$ |
| Two identical rows/columns | $\|A\| = 0$ |
| Proportional rows/columns | $\|A\| = 0$ |
Minors and Cofactors
Minor $M_{ij}$ = determinant of submatrix after deleting row $i$ and column $j$.
Cofactor:
$$\boxed{C_{ij} = (-1)^{i+j} M_{ij}}$$Sign pattern (checkerboard, $+$ at $(1,1)$):
$$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$$Cofactor expansion — row (fix row $i$, sum over $j$):
$$\boxed{|A| = \sum_{j=1}^{n} a_{ij}\, C_{ij}}$$Column (fix column $j$, sum over $i$):
$$\boxed{|A| = \sum_{i=1}^{n} a_{ij}\, C_{ij}}$$Cross-row vanishing property (key for adjoint):
$$\boxed{\sum_{j=1}^{n} a_{ij}\, C_{kj} = 0 \quad (i \neq k)}$$Properties of Determinants
| # | Property | Formula |
|---|---|---|
| 1 | Transpose | $\|A^T\| = \|A\|$ |
| 2 | Row/column interchange | $R_i \leftrightarrow R_j \Rightarrow \|A\| \to -\|A\|$ |
| 3 | Identical rows/columns | $\|A\| = 0$ |
| 4 | Proportional rows/columns | $R_i = kR_j \Rightarrow \|A\| = 0$ |
| 5 | Zero row/column | $\|A\| = 0$ |
| 6 | Scale one row by $k$ | $\|A\| \to k\|A\|$ |
| 7 | Add multiple of a row | $R_i \to R_i + kR_j \Rightarrow \|A\|$ unchanged |
| 8 | Product | $\|AB\| = \|A\|\,\|B\|$ |
| 9 | Identity / zero | $\|I\| = 1,\;\; \|O\| = 0$ |
| 10 | Inverse | $\|A^{-1}\| = \dfrac{1}{\|A\|}$ |
| 11 | Adjoint | $\|\text{adj}(A)\| = \|A\|^{\,n-1}$ |
Consequences of the product rule:
$$\boxed{|A^n| = |A|^n} \qquad \boxed{|kA| = k^n |A| \;\;(n \times n)}$$Standard Determinant Patterns
Vandermonde:
$$\boxed{\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} = (b-a)(c-a)(c-b)}$$General $n \times n$ Vandermonde: $\displaystyle \prod_{1 \le i < j \le n} (a_j - a_i)$
Cyclic determinant:
$$\begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix} = -(a^3 + b^3 + c^3 - 3abc)$$$$= -(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$$1+x$ pattern:
$$\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{vmatrix} = abc + ab + bc + ca$$$$\begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+x \end{vmatrix} = x^2(x+3)$$Adjoint and Inverse
Singular vs non-singular:
$$\boxed{A^{-1} \text{ exists} \iff |A| \neq 0 \;(\text{non-singular})}$$Adjoint = transpose of the cofactor matrix:
$$\boxed{\text{adj}(A) = [C_{ij}]^T}$$2×2 shortcut (swap diagonal, negate off-diagonal):
$$\boxed{\text{adj}\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}$$Inverse formula:
$$\boxed{A^{-1} = \frac{1}{|A|}\,\text{adj}(A)}$$2×2 inverse:
$$\boxed{\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}$$Adjoint Properties ($n \times n$)
| Property | Formula |
|---|---|
| Magic product | $A\cdot\text{adj}(A) = \text{adj}(A)\cdot A = \|A\|\, I$ |
| Determinant of adjoint | $\|\text{adj}(A)\| = \|A\|^{\,n-1}$ |
| Adjoint of transpose | $\text{adj}(A^T) = [\text{adj}(A)]^T$ |
| Adjoint of product | $\text{adj}(AB) = \text{adj}(B)\cdot\text{adj}(A)$ |
| Adjoint of adjoint | $\text{adj}(\text{adj}(A)) = \|A\|^{\,n-2} A$ |
| Adjoint of scalar multiple | $\text{adj}(kA) = k^{\,n-1}\text{adj}(A)$ |
Inverse Properties
| Property | Formula |
|---|---|
| Inverse of inverse | $(A^{-1})^{-1} = A$ |
| Inverse of product (reverses) | $(AB)^{-1} = B^{-1}A^{-1}$ |
| Inverse of transpose | $(A^T)^{-1} = (A^{-1})^T$ |
| Determinant of inverse | $\|A^{-1}\| = \dfrac{1}{\|A\|}$ |
| Inverse of scalar multiple | $(kA)^{-1} = \dfrac{1}{k}A^{-1}$ |
| Identity | $I^{-1} = I$ |
| Diagonal matrix | reciprocal of each diagonal element |
| Orthogonal matrix | $A^{-1} = A^T$ |
| Involutory matrix | $A^{-1} = A$ (since $A^2 = I$) |
Linear Equations
Matrix form of a system: $\boxed{AX = B}$ ($A$ = coefficient matrix, $X$ = variables, $B$ = constants).
Matrix method (when $|A| \neq 0$):
$$\boxed{X = A^{-1}B}$$Cramer’s rule (when $D = |A| \neq 0$):
$$\boxed{x_i = \frac{D_i}{D}}$$where $D_i$ replaces the $i$-th column of $A$ with $B$. For 3 variables:
$$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}$$Consistency of a Square System $AX = B$
| Condition | Nature | Solutions |
|---|---|---|
| $\|A\| \neq 0$ | Consistent | Unique solution |
| $\|A\| = 0,\;(\text{adj }A)B \neq O$ | Inconsistent | No solution |
| $\|A\| = 0,\;(\text{adj }A)B = O$ | Consistent | Infinitely many solutions |
Homogeneous System $AX = O$
- Always has the trivial solution $X = O$.
- Non-trivial solutions exist $\iff |A| = 0$ (then infinitely many).
Geometry Application
Area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$:
$$\boxed{\text{Area} = \frac{1}{2}\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|}$$If this determinant is $0$, the three points are collinear.