JEE Main 2025-26 Mock Test for Matrices and Determinants

A matrix is a rectangular arrangement of numbers into rows and columns. It is classified mainly by its order: m × n (number of rows × number of columns). Types include row matrix, column matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, and zero matrix.

A determinant is a scalar value that can be computed from a square matrix only. It gives important properties of the matrix, such as whether it is invertible or singular, and is useful in solving equations, calculating areas, and changing of variables.

The determinant of a 2×2 matrix A = [[a, b], [c, d]] is calculated as: det(A) = ad - bc. This formula applies only to square matrices of order 2.

Quick tricks for solving such questions include:

• Recognize patterns like row/column operations for simplification
• Use properties (determinant of a triangular matrix is product of diagonal elements)
• If two rows/columns are identical, the determinant is zero
• Shortcut for adjoint and inverse using minors and cofactors
• For competitive exams, practice elimination and substitution for quickest route

Matrices are used in diverse fields such as engineering, physics, economics, computer graphics, cryptography, and more for solving systems of equations, representing data and transformations, encoding messages, and digital image processing.

Matrix addition: Add corresponding elements. For example: A = [[1,2],[3,4]], B = [[5,6],[7,8]]; A+B = [[6,8],[10,12]].
Matrix multiplication: Multiply row elements with column elements and sum. For A (2×2) and B (2×2), AB = [[1*5+2*7, 1*6+2*8], [3*5+4*7, 3*6+4*8]]

• The determinant of a matrix does not change if rows and columns are interchanged.
• If two rows (or columns) of a matrix are identical, its determinant is zero.
• Multiplying a row (or column) by a scalar multiplies the determinant by that scalar.
• The determinant of a product of matrices is equal to the product of their determinants.

The inverse of a matrix A (written as A^{-1}) exists only for square matrices with non-zero determinant. It is the matrix that, when multiplied with the original, yields the identity matrix: AA^{-1} = I.

Matrices and determinants are used to represent systems of linear equations compactly. Methods like Cramer's Rule, matrix inversion, and row reduction enable efficient solutions, especially for multiple equations and unknowns.

You can find matrices and determinants mock tests for JEE Mains and EAMCET on educational websites such as Vedantu, as well as practice PDFs, online test series, and previous year question banks targeted for these exams.

The order of a matrix (number of rows × number of columns) determines how matrices can be added, multiplied, or used in equations. Operations like multiplication are only defined if the number of columns in the first matrix equals the number of rows in the second.