Determinants Notes for Class 12 & IIT JEE

A determinant is a number linked to a square matrix, and it decides whether the matrix has an inverse and whether a system of equations has a unique solution. These determinants notes for Class 12 explain each rule with its formula and a short example, so you can revise the chapter quickly. For solved exercises, use the NCERT Solutions for Class 12 Maths Chapter 4 Determinants.

What Is a Determinant? Definition and Notation 

A determinant is a unique scalar (real number) associated with every square matrix, that is, a matrix with equal numbers of rows and columns. For a square matrix $A$ it is written as $\det(A)$, $|A|$ or $\Delta$.

In determinants notes for Class 12, the first thing to remember is that a matrix is an array of numbers in brackets, while a determinant is the single value computed from it and is written with straight lines. Students lose marks by treating the two as interchangeable.

• Matrix: an array of numbers, not a number itself
• Determinant: a single scalar value
• Singular matrix: $|A| = 0$, no inverse exists
• Non-singular matrix: $|A| \neq 0$, inverse exists

Key distinction for Class 12 and JEE: A matrix is an array of numbers enclosed in brackets [ ] or ( ). A determinant is the scalar value computed from that matrix and is written with straight lines | |. Students frequently lose marks by treating the two as interchangeable — they are not. A matrix is an object; a determinant is a number.

Determinants Notes for Class 12 & IIT JEE

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

India's Best Exam Preparation for Class 12th - Download Now

How to Evaluate a Determinant

Evaluation is the first skill in determinants notes for Class 12, so learn the 2×2 and 3×3 methods before the properties.

Determinant of a 2×2 Matrix

$$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$

Determinant of a 3×3 Matrix by Expansion
Expanding along the first row:

$$\Delta = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$$

You can expand along any row or column. Choose the one with the most zeros to save time.

Example:

$$\begin{vmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{vmatrix} = 1(24 - 0) - 2(0 - 5) + 3(0 - 4) = 22$$

Minors and Cofactors

The minor $M_{ij}$ of the element $a_{ij}$ is the determinant left after deleting row $i$ and column $j$. The cofactor is the signed minor:

$$A_{ij} = (-1)^{i+j}M_{ij}$$

The determinant equals the sum of each element of any row (or column) times its cofactor, so $\Delta = a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3}$. If you multiply the elements of one row by the cofactors of a different row, the sum is zero.

Properties of Determinants for Class 12 and JEE

These properties are the core of determinants notes for Class 12 and save time in long expansions.

• Transposing does not change the value: $|A^T| = |A|$.
• Swapping two rows or two columns changes the sign.
• Two identical rows or columns give zero. A zero row or column, or two proportional rows or columns, also give zero.
• Multiplying one row by $k$ multiplies the determinant by $k$.
• If every element of a row is a sum of two terms, the determinant splits into a sum of two determinants.
• Adding a multiple of one row to another row does not change the value: $R_i \to R_i + kR_j$.
• For a triangular matrix, the determinant is the product of the diagonal elements.
• $|AB| = |A||B|$.
• For an $n \times n$ matrix, $|kA| = k^n|A|$. For example, if $|A| = 4$ for a 3×3 matrix, then $|2A| = 2^3 \times 4 = 32$.

In JEE problems, use row and column operations to create zeros and take out common factors before expanding.

Area of a Triangle Using Determinants

For vertices $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$:

$$\text{Area} = \dfrac{1}{2}\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|$$

If this determinant is zero, the three points are collinear. Example: for $(0,0)$, $(4,0)$ and $(0,3)$ the determinant is 12, so the area is 6 square units.

Adjoint and Inverse of a Matrix

The adjoint of $A$ is the transpose of the matrix of cofactors. Then:

$$A(\text{adj}\,A) = (\text{adj}\,A)A = |A|\,I$$ $$A^{-1} = \dfrac{\text{adj}\,A}{|A|}, \qquad |A| \neq 0$$

Useful results for an $n \times n$ matrix:

$$|\text{adj}\,A| = |A|^{n-1}, \qquad |A^{-1}| = \dfrac{1}{|A|}$$ $$(AB)^{-1} = B^{-1}A^{-1}, \qquad \text{adj}(AB) = (\text{adj}\,B)(\text{adj}\,A)$$

Solving Linear Equations: Cramer’s Rule and Consistency

Cramer’s Rule for Three Equations

For $AX = B$ in three unknowns, let $D = |A|$, and let $D_1$, $D_2$, $D_3$ be $D$ with the first, second and third column replaced by $B$. If $D \neq 0$:

$$x = \dfrac{D_1}{D}, \qquad y = \dfrac{D_2}{D}, \qquad z = \dfrac{D_3}{D}$$

If $D = 0$ and at least one of $D_1$, $D_2$, $D_3$ is non-zero, the system has no solution. If $D = 0$ and all of them are zero, check further, because the system may have infinitely many solutions or none.

Example: for $x + y = 5$ and $x - y = 1$, $D = -2$, $D_1 = -6$, $D_2 = -4$, so $x = 3$ and $y = 2$.

Matrix Method for $AX = B$

If $|A| \neq 0$, the unique solution is $X = A^{-1}B$. If $|A| = 0$ and $(\text{adj}\,A)B \neq O$, there is no solution. If $|A| = 0$ and $(\text{adj}\,A)B = O$, the system is either consistent with infinitely many solutions or inconsistent.

Common Mistakes in Determinants Notes for Class 12

• Treating a matrix and its determinant as the same thing.
• Forgetting the sign pattern $(-1)^{i+j}$ while expanding.
• Using $|kA| = k|A|$ instead of $k^n|A|$.
• Using Cramer’s rule when $D = 0$.

How to Revise Determinants Notes for Class 12

• Practise evaluating 3×3 determinants with row operations until the steps are routine.
• Write the properties and the adjoint and inverse results on one page.
• Solve the exercises and compare your steps with the NCERT Solutions for Class 12 Maths Chapter 4.