Matrices and Determinants — Previous Year Questions (Class 12 Mathematics)
Matrices and determinants are fundamental tools in linear algebra. They are used to solve systems of equations and in various applications across mathemati
TL;DR: Matrices and determinants are fundamental tools in linear algebra. They are used to solve systems of equations and in various applications across math…
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Matrices and determinants are fundamental tools in linear algebra. They are used to solve systems of equations and in various applications across mathemati
Matrices and Determinants — Previous Year Questions with Solutions
Q (2023, 2 marks): Define a matrix and determinant. Explain the difference between a 2×2 and 3×3 determinant.
Answer: Matrix: A rectangular array of numbers arranged in rows and columns.
Example: A = [a b; c d] is a 2×2 matrix
Determinant: A scalar value associated with a square matrix.
For 2×2 matrix: det(A) = ad - bc
For 3×3 matrix: det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)
For matrix A = [a b c; d e f; g h i]
det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)
Differences:
2×2: Single scalar product difference (ad - bc)
3×3: Uses cofactor expansion or Sarrus' rule
2×2: Easier to calculate
3×3: More complex, recursive nature
Q (2022, 1 mark): Calculate the determinant of the matrix A = [2 3; 4 5].
Answer: det(A) = ad - bc
where a=2, b=3, c=4, d=5
det(A) = (2)(5) - (3)(4)
det(A) = 10 - 12
det(A) = -2
Q (2023, 3 marks): Find the inverse of matrix A = [1 2; 3 4]. Verify your answer by multiplying A × A⁻¹.
Answer: For 2×2 matrix A = [a b; c d], inverse is:
A⁻¹ = (1/det(A)) [d -b; -c a]
For A = [1 2; 3 4]:
det(A) = (1)(4) - (2)(3) = 4 - 6 = -2
A⁻¹ = (1/-2) [4 -2; -3 1]
A⁻¹ = [-2 1; 1.5 -0.5]
Or A⁻¹ = [-2 1; 3/2 -1/2]
Verification: A × A⁻¹ = I
[1 2; 3 4] × [-2 1; 3/2 -1/2]
= [1(-2) + 2(3/2) 1(1) + 2(-1/2); 3(-2) + 4(3/2) 3(1) + 4(-1/2)]
= [-2 + 3 1 - 1; -6 + 6 3 - 2]
= [1 0; 0 1] = I ✓
Q (2022, 3 marks): Solve the system of equations using matrix method: 2x + 3y = 8 3x + 4y = 11
Answer: In matrix form: AX = B
[2 3; 3 4] [x; y] = [8; 11]
Solving: X = A⁻¹B
Step 1: Find det(A) = (2)(4) - (3)(3) = 8 - 9 = -1
Step 2: Find A⁻¹ = (1/-1) [4 -3; -3 2] = [-4 3; 3 -2]
Step 3: Multiply X = A⁻¹B
X = [-4 3; 3 -2] [8; 11]
[x; y] = [(-4)(8) + (3)(11); (3)(8) + (-2)(11)]
[x; y] = [-32 + 33; 24 - 22]
[x; y] = [1; 2]
Therefore, x = 1 and y = 2
Q (2023, 3 marks): Calculate the determinant of the 3×3 matrix A = [1 2 3; 0 4 5; 1 0 6].
Answer: Using cofactor expansion along the first row:
det(A) = a11(A11) - a12(A12) + a13(A13)
where:
A11 = det([4 5; 0 6]) = (4)(6) - (5)(0) = 24
A12 = det([0 5; 1 6]) = (0)(6) - (5)(1) = -5
A13 = det([0 4; 1 0]) = (0)(0) - (4)(1) = -4
det(A) = (1)(24) - (2)(-5) + (3)(-4)
det(A) = 24 + 10 - 12
det(A) = 22
Q (2021, 3 marks): If A = [1 2; 3 4] and B = [5 6; 7 8], find AB and BA. Verify whether matrix multiplication is commutative.
Answer: Computing AB:
AB = [1 2; 3 4] × [5 6; 7 8]
AB = [(1)(5)+(2)(7) (1)(6)+(2)(8); (3)(5)+(4)(7) (3)(6)+(4)(8)]
AB = [5+14 6+16; 15+28 18+32]
AB = [19 22; 43 50]
Computing BA:
BA = [5 6; 7 8] × [1 2; 3 4]
BA = [(5)(1)+(6)(3) (5)(2)+(6)(4); (7)(1)+(8)(3) (7)(2)+(8)(4)]
BA = [5+18 10+24; 7+24 14+32]
BA = [23 34; 31 46]
Since AB ≠ BA, matrix multiplication is NOT commutative.
This is a key property of matrix multiplication that distinguishes it from scalar multiplication.
Frequently Asked Questions
What is the condition for a matrix to have an inverse?
A square matrix A has an inverse if and only if its determinant is non-zero (det(A) ≠ 0). Such matrices are called invertible or non-singular. If det(A) = 0, the matrix is singular and does not have an inverse. Geometrically, a singular matrix represents a transformation that reduces dimensionality (like projecting onto a line or point).
What does a determinant tell us geometrically?
For a 2×2 matrix, the determinant represents the area of the parallelogram formed by the row (or column) vectors. For a 3×3 matrix, it represents the volume of the parallelepiped. A positive determinant indicates the matrix preserves orientation, while negative indicates a reversal of orientation. A determinant of zero means the vectors are linearly dependent and the transformation collapses dimensionality.
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