Mathematics Notes for Class 12 - Chapter 4: Determinants

1. Introduction
A determinant is a scalar value that is a function of the entries of a square matrix. It is a number associated with a square matrix. It is essential for solving systems of linear equations, calculating inverse of matrices, and in calculus (Jacobians). We denote the determinant of a square matrix A by det(A) or |A|.

2. Determinant of a Matrix of Order One
Let A = [a] be a matrix of order 1. Then the determinant of A is defined as:

3. Determinant of a Matrix of Order Two
Let A =
[ a11 a12 ]
[ a21 a22 ]
be a matrix of order 2. The determinant of A is defined as:

4. Determinant of a Matrix of Order Three
The determinant of a 3x3 matrix can be calculated using the Sarrus' rule or, more formally, by expansion along a row or column (Laplace expansion).

For matrix A =
[ a11 a12 a13 ]
[ a21 a22 a23 ]
[ a31 a32 a33 ]

The determinant can be expanded along the first row R1 as:

5. Minors and Cofactors

Minor (Mij): The minor of an element aij is the determinant of the matrix obtained by deleting the ith row and the jth column.

Cofactor (Aij): The cofactor of an element aij is defined as Aij = (-1)^(i+j) * Mij.

6. Properties of Determinants

These properties help in simplifying the calculation of determinants:

• Property 1 (Reflection): The value of the determinant remains unchanged if its rows and columns are interchanged (|A| = |A'|).
• Property 2 (Switching): If any two rows (or columns) of a determinant are interchanged, the sign of the determinant changes.
• Property 3 (Repetition): If any two rows (or columns) of a determinant are identical (all corresponding elements are same), the value of the determinant is zero.
• Property 4 (Scalar Multiple): If each element of a row (or a column) of a determinant is multiplied by a constant k, the determinant value is multiplied by k.
• Property 5 (Proportionality): If elements of a row (or column) are proportional to (or identical to) elements of another row (or column), the determinant is zero.
• Property 6 (Sum): If some or all elements of a row or column can be expressed as the sum of two or more terms, the determinant can be expressed as the sum of two or more determinants.
• Property 7 (Elementary Operations): If to each element of any row or column of a determinant, the equimultiples of corresponding elements of one (or more) row (or column) are added, the value of the determinant remains unchanged. This is the key property used to simplify matrices.

7. Area of a Triangle

The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by:

8. Adjoint and Inverse of a Matrix

Adjoint of a Matrix: The adjoint of a square matrix A is the transpose of the cofactor matrix of A.

Theorem 1: For any square matrix A of order n, A * (adj A) = (adj A) * A = |A| * I.

Theorem 2 (Inverse Existence): A square matrix A is invertible (non-singular) if and only if |A| 0.

Inverse Formula: A^-1 = (1 / |A|) * (adj A).

9. Applications of Determinants and Matrices

Determinants are primarily used to solve systems of linear equations. Consider the system:

• a1x + b1y + c1z = d1
• a2x + b2y + c2z = d2
• a3x + b3y + c3z = d3

In matrix form: AX = B, where:
A = Coefficient Matrix
X = Variable Matrix (Column)
B = Constant Matrix (Column)

Consistency of Systems

Case 1 (Unique Solution): If |A| 0, the system is consistent and has a unique solution given by X = A^-1B.

Case 2 (Infinite Solutions or No Solution): If |A| = 0, the system may either have no solution or infinitely many solutions. We must check (adj A) * B in this case.

• If (adj A) * B 0 (Zero matrix), the system is inconsistent (no solution).
• If (adj A) * B = 0 (Zero matrix), the system is consistent and has infinitely many solutions.

10. Summary of Key Points

• Determinant is defined only for square matrices.
• Expanding along a row or column with maximum zeros simplifies calculation.
• Elementary operations (Ri Ri + kRj) leave the |A| unchanged and are vital for simplification.
• A matrix is non-singular iff |A| 0.
• Cramer's Rule is another method to solve equations using determinants for individual variables, though the matrix method (X = A^-1B) is generally preferred.