What is Determinants Class 12: Definition, Properties & Examples

Definition of Determinants in Class 12 Mathematics

In Class 12 NCERT Mathematics, a determinant is a scalar value associated with a square matrix. It provides important information about the matrix, such as whether it is invertible or singular.

For a square matrix $A$ of order $n \times n$, the determinant is denoted by $|A|$ or $\det(A)$.

Example: Determinant of a 2x2 matrix

For matrix

$$ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, $$

the determinant is

$$ |A| = ad - bc. $$

This simple formula forms the basis for understanding determinants of higher order matrices.

Properties of Determinants Every Class 12 Student Should Know

Understanding the properties of determinants helps simplify calculations and solve problems efficiently. Here are the key properties:

• Property 1: The determinant of the identity matrix is 1.
• Property 2: Interchanging two rows (or columns) changes the sign of the determinant.
• Property 3: If two rows (or columns) are identical, the determinant is zero.
• Property 4: Multiplying a row by a scalar multiplies the determinant by the same scalar.
• Property 5: Adding a multiple of one row to another row does not change the determinant.
• Property 6: The determinant of a product of matrices equals the product of their determinants, i.e., $\det(AB) = \det(A) \times \det(B)$.

These properties are crucial for simplifying determinants and understanding matrix behavior.

How to Calculate Determinants of 3x3 Matrices: Step-by-Step Guide

Calculating the determinant of a 3x3 matrix involves expansion by minors or the rule of Sarrus.

Given matrix

$$ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}, $$

its determinant is:

$$ |A| = a(ei - fh) - b(di - fg) + c(dh - eg). $$

Step-by-step example:

Calculate the determinant of

$$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}. $$

• Calculate minors:
• $ei - fh = 5 \times 9 - 6 \times 8 = 45 - 48 = -3$
• $di - fg = 4 \times 9 - 6 \times 7 = 36 - 42 = -6$
• $dh - eg = 4 \times 8 - 5 \times 7 = 32 - 35 = -3$

• Substitute:

$$|A| = 1 \times (-3) - 2 \times (-6) + 3 \times (-3) = -3 + 12 - 9 = 0.$$

Applications of Determinants in Class 12 NCERT Mathematics

Determinants have several important applications in Class 12 Mathematics, including:

• Solving system of linear equations: Using Cramer's Rule, determinants help find solutions when the coefficient matrix is invertible.
• Matrix invertibility: A matrix is invertible if and only if its determinant is non-zero.
• Area and volume calculations: Determinants can calculate the area of parallelograms and volume of parallelepipeds formed by vectors.
• Eigenvalues and eigenvectors: Determinants are used in characteristic equations to find eigenvalues.

These applications make determinants a fundamental topic for exams and further studies.

Comparison Table: Determinants of 2x2 vs 3x3 Matrices

This table helps students quickly differentiate the two common determinant calculations.

Tips and Tricks to Remember Determinants for Class 12 Exams

• Always check if the matrix is square before calculating the determinant.
• Use properties like row operations to simplify determinants before expanding.
• Memorize the 2x2 determinant formula thoroughly.
• Practice expansion by minors for 3x3 matrices regularly.
• Remember that if any row or column has all zeros, the determinant is zero.
• Use determinant properties to avoid lengthy calculations.

Consistent practice with these tips will help you master determinants and score well in exams.