Matrices and Determinants: Basics with Examples
Matrices and determinants are a scoring topic in Class 12 and appear frequently in JEE. They give us a compact way to handle large amounts of numbers and to solve systems of equations. This beginner-friendly guide covers order, types, basic operations, and how to compute a 2x2 determinant with clear examples.
What is a matrix?
A matrix is a rectangular arrangement of numbers in rows and columns. The order of a matrix is written as rows × columns. For example, the matrix [[2, 4, 1], [0, 5, 3]] has 2 rows and 3 columns, so its order is 2 × 3. Each number inside is called an element.
Common types of matrices
- Row matrix: has a single row, e.g. [3 7 2].
- Column matrix: has a single column.
- Square matrix: equal rows and columns, e.g. order 2 × 2 or 3 × 3.
- Diagonal matrix: a square matrix where all non-diagonal elements are 0.
- Identity matrix (I): a diagonal matrix with 1s on the diagonal.
- Zero matrix: every element is 0.
Addition and subtraction of matrices
You can add or subtract two matrices only if they have the same order. You simply add or subtract the elements in matching positions.
Example: Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Then A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]. Similarly, B − A = [[4, 4], [4, 4]].
Matrix multiplication
To multiply matrices, the number of columns in the first must equal the number of rows in the second. Each element of the product is a "row times column" sum.
Example: Multiply A = [[1, 2], [3, 4]] by B = [[5, 6], [7, 8]]. The top-left element is (1)(5) + (2)(7) = 5 + 14 = 19. Completing all four: AB = [[19, 22], [43, 50]]. Note that matrix multiplication is not commutative, so AB is usually not equal to BA. These operations are core to Class 12 Maths and to JEE Maths.
Determinant of a 2x2 matrix
A determinant is a single number associated with a square matrix. For a 2 × 2 matrix A = [[a, b], [c, d]], the determinant is det(A) = ad − bc.
Example: For A = [[3, 8], [4, 6]], det(A) = (3)(6) − (8)(4) = 18 − 32 = −14. If a determinant equals 0, the matrix is called singular and has no inverse; if it is non-zero, the matrix is non-singular and invertible.
Determinants are used to solve simultaneous equations through Cramer's rule and to find inverses. If you find it hard to remember the ad − bc pattern and other rules, our guide on how to memorise maths formulas can help you retain them for the board exam.
Frequently asked questions
What is the difference between a matrix and a determinant?
A matrix is an array of numbers and has no single numerical value, while a determinant is one number calculated from a square matrix. Only square matrices have determinants.
Why is matrix multiplication not commutative?
Because the product depends on multiplying rows of the first matrix by columns of the second, changing the order changes which rows meet which columns. So AB and BA usually give different results.
What does a determinant of zero mean?
A zero determinant means the matrix is singular, so it has no inverse. In systems of equations, it usually signals that there is no unique solution.