

What is Matrix Multiplication in Maths?
The concept of matrix multiplication plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you're studying for boards or preparing for competitive exams like JEE, knowing how to multiply matrices accurately will make solving linear equations, transformations, and data calculations far easier.
What Is Matrix Multiplication?
Matrix multiplication is a special mathematical operation where two matrices are combined to produce a new matrix by multiplying rows of the first matrix with columns of the second. This process is guided by specific rules based on matrix size and order. You’ll find this concept applied in areas such as algebra, computer science, and physics.
Key Formula for Matrix Multiplication
Here’s the standard formula: If \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times p \) matrix, then their product \( AB \) is an \( m \times p \) matrix. Each entry in the product is found by:
\( (AB)_{ij} = \sum_{k=1}^{n} A_{ik} \times B_{kj} \)
This means: multiply elements across the row of the first matrix with the matching column of the second and add up the results.
Cross-Disciplinary Usage
Matrix multiplication is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. It can represent transformations, solve systems of equations, model population growth, and is fundamental for programming and algorithms. Students preparing for JEE, NEET, or CBSE boards will see its relevance in various questions — especially those involving vectors, transformations, and advanced algebra.
Rules and Conditions for Matrix Multiplication
Not all matrices can be multiplied. Remember these two simple rules:
- The number of columns in the first matrix must equal the number of rows in the second matrix.
- The order of the product matrix will be (Rows of first matrix) × (Columns of second matrix).
For example, you can multiply a 2×3 matrix with a 3×2 matrix, but not a 2×3 with a 2×2 matrix.
Step-by-Step Illustration
- Suppose:
\( A = \begin{bmatrix} 3 & 5 \\ 1 & 4 \end{bmatrix} \), \( B = \begin{bmatrix} 2 & 6 \\ 7 & 1 \end{bmatrix} \) - Multiply row 1 of A with column 1 of B:
\( (3 \times 2) + (5 \times 7) = 6 + 35 = 41 \) - Multiply row 1 of A with column 2 of B:
\( (3 \times 6) + (5 \times 1) = 18 + 5 = 23 \) - Multiply row 2 of A with column 1 of B:
\( (1 \times 2) + (4 \times 7) = 2 + 28 = 30 \) - Multiply row 2 of A with column 2 of B:
\( (1 \times 6) + (4 \times 1) = 6 + 4 = 10 \) - So, product matrix \( AB = \begin{bmatrix} 41 & 23 \\ 30 & 10 \end{bmatrix} \)
Common Types and Properties of Matrix Multiplication
Rule / Property | Applies To |
---|---|
Associative Law: (AB)C = A(BC) | All compatible matrices |
Distributive Law: A(B+C) = AB + AC | All compatible matrices |
Matrix multiplication is not commutative: AB ≠ BA in general | General, unless both matrices are square and commute |
Multiplying by identity matrix: AI = IA = A | A is any square matrix |
Speed Trick or Vedic Shortcut
When practicing matrix multiplication, some patterns help save time, especially in MCQ-based exams:
- If multiplying by an identity matrix, the result is the same matrix.
- For diagonal or triangular matrices, you only need to multiply along the diagonals or in upper/lower triangle, reducing calculations.
- Spotting zero-rows or columns helps quickly identify zero elements in the product.
Tricks like these are often covered in Vedantu’s live classes to build your speed and accuracy for board and competitive exams.
Try These Yourself
- Multiply \( \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \) and \( \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \).
- Can you multiply a 2×3 matrix by a 3×1 matrix? What will be the size of the result?
- Solve: Multiply \( \begin{bmatrix}2 \\ -1\end{bmatrix} \) with \( \begin{bmatrix}4 & 7\end{bmatrix} \). What’s the order of the product?
- What is the result if you multiply any matrix with a zero matrix?
Frequent Errors and Misunderstandings
- Attempting to multiply two matrices without checking dimensional compatibility.
- Adding, not multiplying, row and column elements.
- Assuming AB = BA for all matrices (commutative property does not usually hold).
- Miscalculating the order of the resulting matrix.
Relation to Other Concepts
The idea of matrix multiplication connects closely with topics such as matrix addition, matrix inverse, and determinants. Mastering matrix product operations helps you solve systems of linear equations, transformations, and even certain calculus problems.
Classroom Tip
A simple way to remember matrix multiplication: “Row of the first × Column of the second = Entry in the product.” Write small arrows linking each row to each column while practicing. Vedantu’s teachers often use colored highlighters or visual board animations for this in live classes.
We explored matrix multiplication—from its definition, formula, compatible orders, solved examples, shortcuts, and links to other areas of Maths. Continue practicing and review topics like matrices, types of matrices, and elementary operations to strengthen your understanding. The more you try, the easier and faster matrix multiplication becomes in both class and exams!
FAQs on Matrix Multiplication Explained: Steps, Rules & Examples
1. What is matrix multiplication in simple terms?
Matrix multiplication is a mathematical operation that combines two matrices to produce a third, new matrix. The process involves taking the dot product of the rows of the first matrix with the columns of the second matrix. Unlike simple element-wise multiplication, this method is essential for solving systems of linear equations and representing transformations.
2. What is the single most important rule for multiplying two matrices?
The fundamental rule for matrix multiplication is that the number of columns in the first matrix must be equal to the number of rows in the second matrix. For instance, a matrix of order m × n can only be multiplied by a matrix of order n × p. The resulting product matrix will have the order m × p.
3. How is a single element in the final product matrix calculated?
Each element in the product matrix is the result of a row-column calculation. To find the element in the i-th row and j-th column of the product matrix, you multiply each element of the i-th row of the first matrix by the corresponding element of the j-th column of the second matrix, and then sum up all these products. This sum gives you the final value for that specific position.
4. What is the difference between scalar multiplication and matrix multiplication?
The primary difference lies in what is being multiplied.
- Scalar multiplication involves multiplying a single number (a scalar) by every element within a matrix. The matrix's dimensions do not change.
- Matrix multiplication involves multiplying one matrix by another matrix using the row-by-column dot product rule. This can change the dimensions of the resulting matrix and is a more complex operation.
5. Why is matrix multiplication not commutative (i.e., AB ≠ BA)?
Matrix multiplication is generally not commutative for two main reasons. Firstly, the product BA may not even be defined if the dimensions do not align, even if AB is defined. Secondly, even when both products are defined (for example, with two square matrices), the row-by-column multiplication process typically yields different results depending on which matrix comes first. The order of multiplication fundamentally changes the calculation and the final product.
6. What are the key properties of matrix multiplication a student must know?
Besides being non-commutative, matrix multiplication has several important properties as per the CBSE syllabus:
- Associative Law: For any three matrices A, B, and C, (AB)C = A(BC), provided the products are defined.
- Distributive Law: A(B + C) = AB + AC, and (A + B)C = AC + BC, provided the operations are defined.
- Existence of Multiplicative Identity: For every square matrix A, there is an identity matrix I of the same order such that AI = IA = A.
7. How is matrix multiplication used to solve real-world problems?
Matrix multiplication is a powerful tool with many applications. In the context of the Class 12 syllabus, its most important application is in solving systems of linear equations. It is also fundamental in computer graphics for transformations like rotation and scaling, in physics for quantum mechanics, and in economics for input-output models. This shows its importance beyond just algebraic manipulation.
8. What is the role of the identity matrix and the zero matrix in multiplication?
The identity matrix (I) and the zero matrix (O) play special roles in matrix multiplication, similar to the numbers 1 and 0 in regular arithmetic.
- The identity matrix, when multiplied by any compatible matrix A, leaves A unchanged (AI = IA = A). It acts as the multiplicative identity.
- The zero matrix, when multiplied by any compatible matrix A, results in a zero matrix (AO = O or OA = O). It nullifies the matrix it is multiplied with.
9. What is a common conceptual mistake students make when multiplying matrices?
A frequent error, beyond getting the dimensions wrong, is to mistakenly perform element-wise multiplication instead of the correct row-by-column dot product method. For example, a student might multiply the element in the first row, first column of matrix A with the element in the first row, first column of matrix B. The correct method requires multiplying the entire first row of A by the entire first column of B to get a single element.

















