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Matrices and Determinants Formulas For JEE 2026
Matrices and Determinants is an important chapter in JEE Mathematics and usually carries 2–3 questions every year. In this topic, students learn about different types of matrices, matrix operations, transpose, and inverse of a matrix. These basics are very useful for solving algebraic problems and help build confidence in handling matrix-based questions.
The chapter also includes key concepts like determinant properties, adjoint, Cramer’s rule, and the consistency of linear equations, which are often asked in exams. Since many problems are based on applying formulas, regular practice is essential. With proper revision, this chapter can become a scoring area. For quick revision, students can also use a well-organized JEE Mains Maths Formula PDF to go through important formulas and concepts easily.
Matrix
Basic Definitions
A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with $$m$$ rows and $$n$$ columns is called an $$m \times n$$ matrix.
Matrix Notation
$$$A = [a_{ij}]_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$$
- $$a_{ij}$$ = element at row $$i$$, column $$j$$
- The order of the matrix is $$m \times n$$
Types of Matrices
Special Types of Matrices
| Type | Definition |
|---|---|
| Row matrix | $$1 \times n$$ (single row) |
| Column matrix | $$m \times 1$$ (single column) |
| Square matrix | $$m = n$$ (rows = columns) |
| Zero matrix ($$O$$) | All elements are 0 |
| Diagonal matrix | $$a_{ij} = 0$$ for $$i \neq j$$ |
| Scalar matrix | Diagonal with all equal entries |
| Identity matrix ($$I$$) | Diagonal with all 1s |
Identity Matrix: A square matrix $$I_n$$ with 1s on the main diagonal and 0s elsewhere. It acts like the number 1: $$AI = IA = A$$ for any compatible matrix $$A$$.
Triangular Matrices
- Upper triangular: All elements below the main diagonal are zero ($$a_{ij} = 0$$ for $$i > j$$).
- Lower triangular: All elements above the main diagonal are zero ($$a_{ij} = 0$$ for $$i < j$$).
Matrix Operations Formulas
Addition and Scalar Multiplication
If $$A = [a_{ij}]$$ and $$B = [b_{ij}]$$ are both $$m \times n$$ matrices:
$$$A + B = [a_{ij} + b_{ij}]_{m \times n}$$$
If $$k$$ is a scalar: $$kA = [k \cdot a_{ij}]$$ (every element is multiplied by $$k$$).
Matrix Multiplication Formula
If $$A$$ is $$m \times p$$ and $$B$$ is $$p \times n$$, then $$AB$$ is $$m \times n$$ with:
$$$(AB)_{ij} = \sum_{k=1}^{p} a_{ik} \cdot b_{kj}$$$
- $$AB$$ exists only if columns of $$A$$ = rows of $$B$$
- In general, $$AB \neq BA$$ (matrix multiplication is not commutative)
- Matrix multiplication is associative: $$(AB)C = A(BC)$$
- $$AI = IA = A$$ (identity is the multiplicative identity)
Worked Example
Compute $$AB$$ where $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$, $$B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$.
$$(AB)_{11} = 1(5) + 2(7) = 19$$, $$(AB)_{12} = 1(6) + 2(8) = 22$$
$$(AB)_{21} = 3(5) + 4(7) = 43$$, $$(AB)_{22} = 3(6) + 4(8) = 50$$
$$AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$$
Note: $$AB = O$$ does not imply $$A = O$$ or $$B = O$$. Matrix multiplication can give a zero matrix even when neither factor is zero.
Transpose of a Matrix Formulas
Transpose: If $$A = [a_{ij}]_{m \times n}$$, then $$A^T = [a_{ji}]_{n \times m}$$. Rows become columns and columns become rows.
Properties of Transpose
- $$(A^T)^T = A$$
- $$(A + B)^T = A^T + B^T$$
- $$(kA)^T = kA^T$$
- $$(AB)^T = B^T A^T$$ (note the reversal of order!)
Symmetric and Skew-Symmetric Matrices
Symmetric Matrix: A square matrix $$A$$ such that $$A^T = A$$, i.e., $$a_{ij} = a_{ji}$$ for all $$i, j$$.
Skew-Symmetric Matrix: A square matrix $$A$$ such that $$A^T = -A$$, i.e., $$a_{ij} = -a_{ji}$$ for all $$i, j$$. All diagonal elements must be zero.
Key Results
- Every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix: $$$A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$$$
- For a skew-symmetric matrix of odd order, $$\det(A) = 0$$.
Determinants Formula
A determinant is a single number computed from a square matrix. It tells us whether the matrix has an inverse and whether a system of equations has a unique solution.
$$2 \times 2$$ Determinant
$$$\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$$$
$$3 \times 3$$ Determinant (Expansion along Row 1)
$$$\det\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)$$$
Worked Example
Find $$\det(A)$$ where $$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$.
$$= 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = -3 + 12 - 9 =$$ 0
Properties of Determinants
Properties of Determinants
- $$|A^T| = |A|$$ (transpose does not change the determinant)
- If any two rows (or columns) are interchanged, the determinant changes sign.
- If any two rows (or columns) are identical, $$|A| = 0$$.
- If all elements of a row are multiplied by $$k$$, the determinant is multiplied by $$k$$.
- $$|kA| = k^n |A|$$ for an $$n \times n$$ matrix.
- Adding a multiple of one row to another does not change the determinant.
- $$|AB| = |A| \cdot |B|$$
Worked Example
If $$A$$ is a $$3 \times 3$$ matrix and $$|A| = 5$$, find $$|3A|$$.
$$|3A| = 3^3 |A| = 27 \times 5 =$$ 135
Tip: The property "adding a multiple of one row to another doesn't change the determinant" is the most powerful tool for simplification. Use it to create zeros, making expansion easier.
Minors and Cofactors Formulas
Minor: The minor $$M_{ij}$$ of element $$a_{ij}$$ is the determinant of the submatrix obtained by deleting the $$i$$-th row and $$j$$-th column.
Cofactor: The cofactor $$C_{ij}$$ of element $$a_{ij}$$ is: $$C_{ij} = (-1)^{i+j} M_{ij}$$.
Cofactor Expansion
The determinant can be computed along any row or column:
$$$|A| = \sum_{j=1}^{n} a_{ij} C_{ij} \quad \text{(expansion along row } i\text{)}$$$
Note: The sum of elements of one row multiplied by cofactors of a different row is always zero: $$\sum_{j=1}^{n} a_{ij} C_{kj} = 0$$ when $$i \neq k$$.
Adjoint of a Matrix Formulas
Adjoint: The adjoint of a square matrix $$A$$, written $$\text{adj}(A)$$, is the transpose of the cofactor matrix: $$\text{adj}(A) = [C_{ij}]^T$$.
Properties of Adjoint
- $$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A| \cdot I$$
- $$|\text{adj}(A)| = |A|^{n-1}$$ for an $$n \times n$$ matrix
- $$\text{adj}(AB) = \text{adj}(B) \cdot \text{adj}(A)$$
- $$\text{adj}(A^T) = (\text{adj}(A))^T$$
Worked Example
Find $$\text{adj}(A)$$ for $$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$$.
For a $$2 \times 2$$ matrix $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, the adjoint is $$\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$.
$$\text{adj}(A) = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$$
Tip: For $$2 \times 2$$ matrices, the adjoint is easy to remember: swap the diagonal elements, and negate the off-diagonal elements.
Inverse of a Matrix Formula
Singular Matrix: A square matrix with $$|A| = 0$$. Singular matrices do not have an inverse.
Non-singular Matrix: A square matrix with $$|A| \neq 0$$. Non-singular matrices have a unique inverse.
Inverse of a Matrix
$$$A^{-1} = \frac{1}{|A|} \text{adj}(A) \qquad (|A| \neq 0)$$$
$$2 \times 2$$ Inverse
For $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ with $$ad - bc \neq 0$$:
$$$A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$$
Properties of Inverse
- $$(A^{-1})^{-1} = A$$
- $$(AB)^{-1} = B^{-1}A^{-1}$$ (reverse order!)
- $$(A^T)^{-1} = (A^{-1})^T$$
- $$|A^{-1}| = \dfrac{1}{|A|}$$
- $$(kA)^{-1} = \dfrac{1}{k}A^{-1}$$
Worked Example
Find $$A^{-1}$$ for $$A = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$$.
$$|A| = 2(3) - 5(1) = 1 \neq 0$$ (inverse exists).
$$A^{-1} = \dfrac{1}{1}\begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix}$$
Verify: $$AA^{-1} = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$$ ✓
Cramer's Rule for Solving Linear Equations
For the system $$AX = B$$ where $$A$$ is $$n \times n$$ and $$|A| \neq 0$$:
$$$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}$$$
- $$D = |A|$$ = determinant of the coefficient matrix
- $$D_x$$ = determinant obtained by replacing the $$x$$-column with $$B$$
- $$D_y$$ = determinant obtained by replacing the $$y$$-column with $$B$$
- $$D_z$$ = determinant obtained by replacing the $$z$$-column with $$B$$
Worked Example
Solve: $$2x + y = 5$$, $$3x + 2y = 8$$.
$$D = \begin{vmatrix} 2 & 1 \\ 3 & 2 \end{vmatrix} = 4 - 3 = 1$$
$$D_x = \begin{vmatrix} 5 & 1 \\ 8 & 2 \end{vmatrix} = 10 - 8 = 2$$, $$D_y = \begin{vmatrix} 2 & 5 \\ 3 & 8 \end{vmatrix} = 16 - 15 = 1$$
$$x = \dfrac{2}{1} =$$ 2, $$y = \dfrac{1}{1} =$$ 1
Matrix Inverse Method
Solving $$AX = B$$ Using the Inverse
If $$|A| \neq 0$$:
$$$X = A^{-1}B$$$
Consistency of Linear Equations
Classification of Systems $$AX = B$$
Let $$D = |A|$$:
Case 1: $$D \neq 0$$ (Non-singular)
- The system has a unique solution (consistent).
Case 2: $$D = 0$$ (Singular)
- If $$(\text{adj}\,A) \cdot B = O$$: infinitely many solutions (consistent).
- If $$(\text{adj}\,A) \cdot B \neq O$$: no solution (inconsistent).
Homogeneous System $$AX = O$$
A homogeneous system always has at least the trivial solution $$X = O$$.
- If $$|A| \neq 0$$: only the trivial solution $$x = y = z = 0$$.
- If $$|A| = 0$$: infinitely many (non-trivial) solutions exist.
Worked Example
For what value of $$k$$ does the system $$x + y + z = 1$$, $$x + 2y + 4z = k$$, $$x + 4y + 10z = k^2$$ have a solution?
$$D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 4 & 10 \end{vmatrix} = 4 - 6 + 2 = 0$$
Since $$D = 0$$, check $$D_x$$:
$$D_x = 2(k-1)(k-2)$$
For consistency: $$D_x = 0 \Rightarrow k = 1$$ or $$k = 2$$.
The system has solutions for $$k =$$ 1 or $$k =$$ 2.
Tip: For JEE, remember: $$D \neq 0$$ always means a unique solution. When $$D = 0$$, you must check further. For a homogeneous system, $$D = 0$$ guarantees non-trivial solutions.
Quick Reference
Key Formulas at a Glance
| Concept | Formula / Result |
|---|---|
| $$2\times 2$$ determinant | $$ad - bc$$ |
| $$|kA|$$ for $$n \times n$$ | $$k^n|A|$$ |
| $$|AB|$$ | $$|A| \cdot |B|$$ |
| Adjoint relation | $$A \cdot \text{adj}(A) = |A| \cdot I$$ |
| Inverse | $$A^{-1} = \frac{1}{|A|}\text{adj}(A)$$ |
| $$(AB)^{-1}$$ | $$B^{-1}A^{-1}$$ |
| $$(AB)^T$$ | $$B^T A^T$$ |
| Cramer's Rule | $$x = D_x/D$$, $$y = D_y/D$$, $$z = D_z/D$$ |
| $$|\text{adj}(A)|$$ | $$|A|^{n-1}$$ |
Matrices and Determinants Formulas For JEE 2026: Conclusion
Matrices and determinants play a crucial role in building a strong foundation in mathematics, especially for competitive exams. With clear understanding of concepts like matrix operations, determinants, inverse, and Cramer’s rule, students can solve a wide range of problems efficiently. Since many questions are formula-based, regular revision and practice are the key to mastering this chapter.
To perform well, students should focus on understanding properties and applying them correctly in different problem types. Consistent practice, along with revising important formulas and shortcuts, can significantly improve speed and accuracy. With the right preparation strategy, this topic can become one of the most scoring areas in the exam.
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