If two rows of a determinant are interchanged, its value:

Changes sign. Interchanging rows changes the sign.

If two rows of a determinant are identical, its value is:

0. A determinant with identical rows is zero.

The value of |1 2 3; 4 5 6; 7 8 9| is:

0. 1(45 − 48) − 2(36 − 42) + 3(32 − 35) = −3 + 12 − 9 = 0.

If every element of one row is multiplied by 3, the determinant is:

Multiplied by 3. Only one row is multiplied, so the value is multiplied by 3.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 6: Determinants

Short notes, key concepts, quick revision and practice MCQs with answers

A determinant is a number attached to a square arrangement of numbers. The chapter expands determinants of order 2 and 3, studies minors, cofactors and the properties that simplify evaluation, and applies determinants to solve linear equations by Cramer's rule, test consistency, find the area of a triangle and check collinearity.

Key concepts: Determinants

Determinant of order 2: |a b; c d| = ad − bc.
Determinant of order 3: Expanded along the first row: a₁(b₂c₃ − b₃c₂) − b₁(a₂c₃ − a₃c₂) + c₁(a₂b₃ − a₃b₂). It can be expanded along any row or column with the sign pattern + − +.
Minor and cofactor: The minor Mᵢⱼ is the determinant left after deleting row i and column j. The cofactor Aᵢⱼ = (−1)^(i+j) Mᵢⱼ.
Properties: Value unchanged when rows and columns are interchanged; sign changes when two rows (columns) are interchanged; zero if two rows (columns) are identical; multiplying a row by k multiplies the value by k; adding k times one row to another leaves the value unchanged.
Triangle property: If all elements above or below the main diagonal are zero, the value is the product of the diagonal elements.
Cramer's rule: For a system of linear equations with D ≠ 0: x = Dx/D, y = Dy/D, z = Dz/D, where Dx, Dy, Dz replace the column of that variable by the constants.
Consistency: Three equations a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0, a₃x + b₃y + c₃ = 0 in two variables are consistent if the determinant of their coefficients and constants is zero.
Area and collinearity: Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) = (1/2)|x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1| (take the absolute value). The points are collinear if this determinant is 0.

Quick revision: important points

A determinant with a row or column of zeros has value 0.
If each element of an n × n determinant is multiplied by k, the value is multiplied by kⁿ.
The sum of products of the elements of a row with their own cofactors gives the value of the determinant.
|1 a a²; 1 b b²; 1 c c²| = (a − b)(b − c)(c − a).
Cramer's rule cannot be used when D = 0.
Area is always taken as positive.

Practice MCQs with answers

The value of |2 3; 1 4| is:
1. 11
2. 5
3. −5
4. 8
Show answer
Answer: B. 5
2 × 4 − 3 × 1 = 5.
If two rows of a determinant are interchanged, its value:
1. Remains the same
2. Becomes zero
3. Changes sign
4. Doubles
Show answer
Answer: C. Changes sign
Interchanging rows changes the sign.
If two rows of a determinant are identical, its value is:
1. 1
2. 0
3. Unchanged
4. Doubled
Show answer
Answer: B. 0
A determinant with identical rows is zero.
The value of |1 2 3; 4 5 6; 7 8 9| is:
1. 0
2. 3
3. −3
4. 6
Show answer
Answer: A. 0
1(45 − 48) − 2(36 − 42) + 3(32 − 35) = −3 + 12 − 9 = 0.
If every element of one row is multiplied by 3, the determinant is:
1. Unchanged
2. Multiplied by 3
3. Multiplied by 9
4. Multiplied by 27
Show answer
Answer: B. Multiplied by 3
Only one row is multiplied, so the value is multiplied by 3.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters