NCERT Solutions Class 12 Maths Chapter 3 provides comprehensive solutions to all 56 questions on Matrices, covering matrix operations, transpose, symmetric and skew-symmetric matrices, and elementary row operations. You’ll learn how to add, multiply, and find inverses of matrices using adjoint and elementary transformations—essential skills for solving systems of linear equations and applications in physics, economics, and computer science. Each solution includes detailed steps to help you tackle board exam problems confidently.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 3 Matrices – Complete Guide
NCERT Class 12 Chapter 3 on Matrices introduces you to one of the most powerful mathematical tools used in engineering, physics, computer graphics, and economics. You’ll explore how matrices serve as compact representations of data and linear transformations, making complex calculations manageable and systematic.
📊 CBSE Class 12 Maths Chapter 3 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 5 Marks |
| Unit Name | Algebra |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
In this chapter, you’ll learn about different types of matrices including row matrices, column matrices, square matrices, diagonal matrices, scalar matrices, identity matrices, and zero matrices. You’ll master fundamental operations such as matrix addition, subtraction, scalar multiplication, and matrix multiplication. Understanding the properties of these operations—like associativity, commutativity (where applicable), and distributivity—will help you manipulate matrices confidently. You’ll also study the transpose of a matrix and symmetric and skew-symmetric matrices, which have important applications in physics and engineering.
The chapter holds medium importance for CBSE board exams with a weightage of approximately 5 marks. You can expect questions ranging from 2-mark problems on basic matrix operations and properties to 4-mark questions involving matrix multiplication and solving matrix equations. The concepts are moderately challenging but become manageable with consistent practice. Matrix operations follow specific rules that differ from regular arithmetic, so paying attention to these distinctions is crucial.
Quick Facts – Class 12 Chapter 3
| 📖 Chapter Number | Chapter 3 |
| 📚 Chapter Name | Matrices |
| ✏️ Total Exercises | 5 Exercises |
| ❓ Total Questions | 56 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering matrices is essential not just for board exams but also for competitive examinations and higher studies in mathematics, engineering, and data science. The systematic approach you develop while working with matrices will enhance your problem-solving abilities across various mathematical domains. With regular practice of NCERT solutions and previous years’ CBSE questions, you’ll build strong conceptual clarity and computational skills in this fundamental topic.
NCERT Solutions Class 12 Maths Chapter 3 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 3.1 | Complete step-by-step solutions for 10 questions | 📥 Download PDF |
| Exercise 3.2 | Complete step-by-step solutions for 22 questions | 📥 Download PDF |
| Exercise 3.3 | Complete step-by-step solutions for 12 questions | 📥 Download PDF |
| Exercise 3.4 | Complete step-by-step solutions for 1 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 3 | Complete step-by-step solutions for 11 questions | 📥 Download PDF |
Matrices – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Matrix Addition \(A + B = [a_{ij} + b_{ij}]\) | Adds corresponding elements of two matrices A and B. Note: Matrices must have the same order (same number of rows and columns) to be added. Remember to add corresponding elements only. | When you need to combine two matrices of the SAME order. |
| Scalar Multiplication \(kA = [ka_{ij}]\) | Multiplies each element of matrix A by a scalar k. Note: Scalar multiplication changes the magnitude of each element in the matrix. Useful for simplifying or scaling matrices. | When you need to multiply a matrix by a constant number. |
| Matrix Multiplication \(C = AB\), where \(c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}\) | Multiplies matrix A (m x n) by matrix B (n x p) to get matrix C (m x p). Note: The number of columns in A MUST equal the number of rows in B. Multiplication is NOT commutative (AB ≠ BA in general). Be careful with the order! | When you need to combine two matrices where the number of columns in the first matrix equals the number of rows in the second matrix. |
| Transpose of a Matrix \( (A^T)_{ij} = a_{ji} \) | Interchanges rows and columns of matrix A. Note: The rows of A become the columns of Aᵀ, and vice-versa. Order changes from m x n to n x m. | When you need to switch rows and columns, often used in symmetric/skew-symmetric matrix problems. |
| Properties of Transpose \((A^T)^T = A, (kA)^T = kA^T, (A+B)^T = A^T + B^T, (AB)^T = B^T A^T\) | Important properties related to transpose operation. Note: Pay special attention to the last property: (AB)ᵀ = BᵀAᵀ, the order is reversed! | Simplifying expressions involving transposes, proving identities. |
| Adjoint of a 2×2 Matrix If \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), then \(adj(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\) | Finds the adjoint of a 2×2 matrix by swapping diagonal elements and negating off-diagonal elements. Note: Only applicable to 2×2 matrices. Remember to swap the diagonal elements AND change the signs of the off-diagonal elements. | Calculating the inverse of a 2×2 matrix quickly. |
| Adjoint of a Matrix (General) \(adj(A) = [C_{ji}]\), where \(C_{ji}\) is the cofactor of \(a_{ij}\) | The adjoint of a matrix is the transpose of the cofactor matrix. Note: Calculate the cofactor of each element, arrange them in a matrix, and then take the transpose. Remember sign conventions for cofactors! | Finding the inverse of a matrix (any order). |
| Inverse of a Matrix \(A^{-1} = \frac{1}{|A|} adj(A)\) | Calculates the inverse of a square matrix A. Note: A matrix is invertible (non-singular) if and only if its determinant |A| is non-zero. If |A| = 0, the inverse does not exist. | Solving matrix equations, finding the solution to linear systems of equations. |
| Properties of Inverse \( (A^{-1})^{-1} = A, (AB)^{-1} = B^{-1}A^{-1}, (A^T)^{-1} = (A^{-1})^T \) | Key properties of the inverse of a matrix. Note: Remember that the inverse of a product is the product of the inverses in the REVERSE order: (AB)⁻¹ = B⁻¹A⁻¹ | Simplifying expressions with inverses, proving matrix identities. |
| Determinant of a 2×2 Matrix If \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), then \(|A| = ad – bc\) | Calculates the determinant of a 2×2 matrix. Note: Easy to calculate, but only works for 2×2 matrices. Remember the order: (a*d) – (b*c). | Checking invertibility, solving linear equations. |
| Determinant of a 3×3 Matrix \(|A| = a_{11}(a_{22}a_{33} – a_{23}a_{32}) – a_{12}(a_{21}a_{33} – a_{23}a_{31}) + a_{13}(a_{21}a_{32} – a_{22}a_{31})\) | Calculates the determinant of a 3×3 matrix. Note: Expand along any row or column, but be careful with the signs (+ – + or – + -). Choose a row/column with zeros to simplify calculation. | Checking invertibility, solving linear equations. |
| Solving System of Linear Equations using Matrix Inverse If \(AX = B\), then \(X = A^{-1}B\) | Solves a system of linear equations represented in matrix form. Note: A must be a square matrix. Make sure your equations are in the correct format (AX = B) before applying the formula. Calculate A⁻¹ first! | Solving systems of equations when the number of equations equals the number of unknowns and the coefficient matrix is invertible. |
Frequently Asked Questions – NCERT Class 12 Maths Chapter 3
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