Determinants

CBSE Class 12 Maths · 11 questions · 26 marks

Every square matrix carries a single number, its determinant, that decides whether the matrix is invertible and measures how it scales area or volume. This chapter builds the determinant from 2×2 up to 3×3 expansion, uses it to test collinearity of points and area of a triangle, and packages the adjoint into a clean formula for A⁻¹ — the algebraic partner to the row-operation method from the previous chapter.

Quick revision

For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc.
A 3×3 determinant is evaluated by expanding along any row or column using cofactors: |A| = Σ aᵢⱼ Cᵢⱼ for a fixed row or column.
Area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) = (1/2)|x1(y2−y3) + x2(y3−y1) + x3(y1−y2)|.
Three points are collinear exactly when the area computed from the determinant formula equals 0.
The adjoint of A, adj(A), is the transpose of the matrix of cofactors of A; it satisfies A·adj(A) = adj(A)·A = |A|·I.
A square matrix A is invertible if and only if |A| ≠ 0; when invertible, A⁻¹ = adj(A)/|A|. A matrix with |A| = 0 is called singular.
For an n×n matrix A and scalar k, |kA| = kⁿ|A|; also |A'| = |A| and |AB| = |A|·|B|.
If any row or column of a determinant is entirely zero, or if two rows/columns are identical, the determinant equals 0.
For an n×n invertible matrix A, |adj(A)| = |A|ⁿ⁻¹.

Important questions

1
MCQ1 mark
The value of the determinant |[3,4],[1,2]| is:
1. A.2
2. B.10
3. C.-2
4. D.14
Answer
The value is 2. For a 2×2 determinant |[a,b],[c,d]| = ad − bc. Here ad − bc = 3×2 − 4×1 = 6 − 4 = 2.
2
MCQ1 mark
If A is a 3×3 matrix with |A| = 5, then |3A| equals:
1. A.15
2. B.135
3. C.45
4. D.125
Answer
|3A| = 135. For an n×n matrix, |kA| = kⁿ|A|. Here n = 3 and k = 3. So |3A| = 3³ × 5 = 27 × 5 = 135.
3
MCQ1 mark
The area of the triangle with vertices (2,3), (-1,0) and (2,-4) is:
1. A.10.5 sq units
2. B.21 sq units
3. C.7 sq units
4. D.14 sq units
Answer
The area is 10.5 sq units. Area = (1/2)|x1(y2−y3) + x2(y3−y1) + x3(y1−y2)|. Substituting: (1/2)|2(0-(-4)) + (-1)(-4-3) + 2(3-0)| = (1/2)|2×4 + (-1)×(-7) + 2×3| = (1/2)|8+7+6|. = (1/2)(21) = 10.5 sq units.
4
MCQ1 mark
If A = [[2,0],[0,2]], then A⁻¹ equals:
1. A.[[1/2,0],[0,1/2]]
2. B.[[2,0],[0,2]]
3. C.[[-1/2,0],[0,-1/2]]
4. D.[[4,0],[0,4]]
Answer
A⁻¹ = [[1/2,0],[0,1/2]]. |A| = 2×2 − 0×0 = 4, and adj(A) = [[2,0],[0,2]] since the off-diagonal cofactors vanish. A⁻¹ = adj(A)/|A| = (1/4)[[2,0],[0,2]] = [[1/2,0],[0,1/2]].
5
Assertion & Reason1 mark
Assertion (A): If two rows of a determinant are identical, the value of the determinant is zero. Reason (R): Interchanging two rows of a determinant negates its value, so a matrix with two identical rows satisfies D = -D.
1. A.Both A and R are true and R is the correct explanation of A
2. B.Both A and R are true but R is not the correct explanation of A
3. C.A is true but R is false
4. D.A is false but R is true
Answer
Both A and R are true and R is the correct explanation of A. Reason R is a standard property: swapping any two rows multiplies the determinant by −1. If the two identical rows are swapped, the determinant D must equal −D (since swapping identical rows changes nothing visibly), forcing 2D = 0, so D = 0. This directly proves Assertion A, so R correctly explains A.
6
Short answer2 marks
Using determinants, check whether the points (1,2), (3,6) and (5,10) are collinear.
7
Short answer2 marks
Find the value of x if |[x,2],[3,x]| = |[4,2],[1,3]|.
8
Short answer3 marks
If A is a 3×3 matrix with |A| = 4, find the value of |adj(A)|.
9
Long answer5 marks
Find the inverse of A = [[2,-1,1],[-1,2,-1],[1,-1,2]] using the adjoint method.
10
Long answer5 marks
Using Cramer's rule (the determinant method), solve the system: x + y + z = 6, y + 3z = 11, x - 2y + z = 0.
11
Case study4 marks
Read the following and answer the questions that follow: Two friends invest in shares of two companies. Matrix A = [[5000,3000],[2000,7000]] gives the amounts invested in rupees, where rows represent Friend 1 and Friend 2, and columns represent Company X and Company Y. (a) Find |A|. (b) Is A invertible? Justify your answer. (c) Find adj(A). (d) Find A⁻¹.

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Determinants

CBSE Class 12 Maths · 11 questions · 26 marks

Quick revision

For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc.

A 3×3 determinant is evaluated by expanding along any row or column using cofactors: |A| = Σ aᵢⱼ Cᵢⱼ for a fixed row or column.

Area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) = (1/2)|x1(y2−y3) + x2(y3−y1) + x3(y1−y2)|.

Three points are collinear exactly when the area computed from the determinant formula equals 0.

The adjoint of A, adj(A), is the transpose of the matrix of cofactors of A; it satisfies A·adj(A) = adj(A)·A = |A|·I.

A square matrix A is invertible if and only if |A| ≠ 0; when invertible, A⁻¹ = adj(A)/|A|. A matrix with |A| = 0 is called singular.

For an n×n matrix A and scalar k, |kA| = kⁿ|A|; also |A'| = |A| and |AB| = |A|·|B|.

If any row or column of a determinant is entirely zero, or if two rows/columns are identical, the determinant equals 0.

For an n×n invertible matrix A, |adj(A)| = |A|ⁿ⁻¹.

Important questions

MCQ1 mark

The value of the determinant |[3,4],[1,2]| is:

A.2
B.10
C.-2
D.14

MCQ1 mark

If A is a 3×3 matrix with |A| = 5, then |3A| equals:

A.15
B.135
C.45
D.125

MCQ1 mark

The area of the triangle with vertices (2,3), (-1,0) and (2,-4) is:

A.10.5 sq units
B.21 sq units
C.7 sq units
D.14 sq units

MCQ1 mark

If A = [[2,0],[0,2]], then A⁻¹ equals:

A.[[1/2,0],[0,1/2]]
B.[[2,0],[0,2]]
C.[[-1/2,0],[0,-1/2]]
D.[[4,0],[0,4]]

Assertion & Reason1 mark

Assertion (A): If two rows of a determinant are identical, the value of the determinant is zero. Reason (R): Interchanging two rows of a determinant negates its value, so a matrix with two identical rows satisfies D = -D.

A.Both A and R are true and R is the correct explanation of A
B.Both A and R are true but R is not the correct explanation of A
C.A is true but R is false
D.A is false but R is true

Short answer2 marks

Using determinants, check whether the points (1,2), (3,6) and (5,10) are collinear.

Short answer2 marks

Find the value of x if |[x,2],[3,x]| = |[4,2],[1,3]|.

Short answer3 marks

If A is a 3×3 matrix with |A| = 4, find the value of |adj(A)|.

Long answer5 marks

Find the inverse of A = [[2,-1,1],[-1,2,-1],[1,-1,2]] using the adjoint method.

Long answer5 marks

Using Cramer's rule (the determinant method), solve the system: x + y + z = 6, y + 3z = 11, x - 2y + z = 0.

Case study4 marks

Read the following and answer the questions that follow: Two friends invest in shares of two companies. Matrix A = [[5000,3000],[2000,7000]] gives the amounts invested in rupees, where rows represent Friend 1 and Friend 2, and columns represent Company X and Company Y. (a) Find |A|. (b) Is A invertible? Justify your answer. (c) Find adj(A). (d) Find A⁻¹.