Matrices — Class 12 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Matrices" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 12 Mathematics chapter "Matrices" — 8 important questions with detailed answers for CBSE board exam prepar…
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NCERT Solutions for Class 12 Mathematics — Matrices. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Matrices — All 8 Questions Solved
Q1. Find x and y if [x+y, 2y] = [7, 8].
Given: [x + y, 2y] = [7, 8]
Two matrices (or arrays) are equal if and only if their corresponding elements are equal.
Comparing corresponding elements:
First element: x + y = 7 ... (1)
Second element: 2y = 8 ... (2)
From equation (2):
2y = 8
y = 4
Substituting y = 4 into equation (1):
x + 4 = 7
x = 3
VERIFICATION:
[x + y, 2y] = [3 + 4, 2(4)] = [7, 8] ✓
CONCLUSION: x = 3 and y = 4
Q2. If A = [1 2; 3 4] and B = [2 0; 1 3], find A + B.
Given: A = [1 2; 3 4] and B = [2 0; 1 3]
To find A + B, add corresponding elements:
A + B = [1 2; 3 4] + [2 0; 1 3]
= [1+2, 2+0; 3+1, 4+3]
= [3, 2; 4, 7]
CONCLUSION: A + B = [3 2; 4 7]
Q3. If A = [1 2 3; 4 5 6] and k = 2, find kA.
Given: A = [1 2 3; 4 5 6] and k = 2
Scalar multiplication of a matrix is performed by multiplying each element by the scalar:
kA = 2 × [1 2 3; 4 5 6]
= [2×1, 2×2, 2×3; 2×4, 2×5, 2×6]
= [2, 4, 6; 8, 10, 12]
CONCLUSION: kA = [2 4 6; 8 10 12]
Q4. If A = [1 2; 3 4], find A² (i.e., A × A).
Given: A = [1 2; 3 4]
We need to find A × A.
For multiplication, the element at position (i,j) in the result is the dot product of row i of the first matrix and column j of the second matrix.
A² = [1 2; 3 4] × [1 2; 3 4]
Element (1,1): (1)(1) + (2)(3) = 1 + 6 = 7
Element (1,2): (1)(2) + (2)(4) = 2 + 8 = 10
Element (2,1): (3)(1) + (4)(3) = 3 + 12 = 15
Element (2,2): (3)(2) + (4)(4) = 6 + 16 = 22
A² = [7, 10; 15, 22]
CONCLUSION: A² = [7 10; 15 22]
Q5. If A = [2 3; 1 4], find the transpose of A (denoted Aᵀ).
Given: A = [2 3; 1 4]
The transpose of a matrix A is obtained by interchanging its rows and columns.
Rows of A: Row 1 = [2, 3], Row 2 = [1, 4]
Columns of A: Column 1 = [2; 1], Column 2 = [3; 4]
Aᵀ (transpose) has columns of A as its rows:
Aᵀ = [2, 1; 3, 4]
Alternatively, element at position (i, j) in Aᵀ equals element at position (j, i) in A.
VERIFICATION:
A₁₁ = 2 → (Aᵀ)₁₁ = 2 ✓
A₁₂ = 3 → (Aᵀ)₂₁ = 3 ✓
A₂₁ = 1 → (Aᵀ)₁₂ = 1 ✓
A₂₂ = 4 → (Aᵀ)₂₂ = 4 ✓
CONCLUSION: Aᵀ = [2 1; 3 4]
Q6. Compute A - 2B if A = [5 3; 2 1] and B = [1 2; 0 3].
Given: A = [5 3; 2 1] and B = [1 2; 0 3]
Step 1: Find 2B
2B = 2 × [1 2; 0 3] = [2, 4; 0, 6]
Step 2: Find A - 2B
A - 2B = [5 3; 2 1] - [2 4; 0 6]
= [5-2, 3-4; 2-0, 1-6]
= [3, -1; 2, -5]
CONCLUSION: A - 2B = [3 -1; 2 -5]
Q7. If [x; y; z] = [2; 3; 5], find the values of x, y, and z.
Given: [x; y; z] = [2; 3; 5]
Two matrices are equal if and only if their corresponding elements are equal.
Comparing corresponding elements:
First element: x = 2
Second element: y = 3
Third element: z = 5
CONCLUSION: x = 2, y = 3, z = 5
Q8. If A = [1 0; 0 1], show that A² = A (the identity matrix squared equals itself).
Given: A = [1 0; 0 1] (the identity matrix I)
We need to compute A² = A × A.
A² = [1 0; 0 1] × [1 0; 0 1]
Element (1,1): (1)(1) + (0)(0) = 1 + 0 = 1
Element (1,2): (1)(0) + (0)(1) = 0 + 0 = 0
Element (2,1): (0)(1) + (1)(0) = 0 + 0 = 0
Element (2,2): (0)(0) + (1)(1) = 0 + 1 = 1
A² = [1 0; 0 1]
Comparing with A = [1 0; 0 1]:
A² = A
CONCLUSION: The identity matrix has the property that I² = I. This is proven. ✓
Matrices — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. If a matrix has 24 elements, what are the possible orders it can have? (How many possible orders exist?)
- A. 4
- B. 6
- C. 8
- D. 10
Answer: C. 8 — The number of elements in a matrix of order m x n is mn. We need to find the number of pairs (m, n) such that mn = 24. The factors of 24 are (1,24), (2,12), (3,8), (4,6), (6,4), (8,3), (12,2), (24,1), which are 8 possible orders.
MCQ 2. Construct a 2x2 matrix A = [a_ij] whose elements are given by a_ij = (i+j)^2 / 2.
- A. [[2, 9/2], [9/2, 8]]
- B. [[1, 2], [3, 4]]
- C. [[2, 5/2], [5/2, 4]]
- D. [[1/2, 2], [2, 9/2]]
Answer: A. [[2, 9/2], [9/2, 8]] — Calculate each element: a_11 = (1+1)^2 / 2 = 2^2 / 2 = 2. a_12 = (1+2)^2 / 2 = 3^2 / 2 = 9/2. a_21 = (2+1)^2 / 2 = 3^2 / 2 = 9/2. a_22 = (2+2)^2 / 2 = 4^2 / 2 = 8. Thus, A = [[2, 9/2], [9/2, 8]].
MCQ 3. Which of the following is a scalar matrix?
- A. [[1, 0], [0, 1]]
- B. [[2, 0], [0, 0]]
- C. [[0, 0], [0, 0]]
- D. [[3, 0], [0, 3]]
Answer: D. [[3, 0], [0, 3]] — A scalar matrix is a diagonal matrix where all diagonal elements are equal. Option '[[3, 0], [0, 3]]' is a diagonal matrix with all diagonal elements equal to 3.
MCQ 4. If A = [[sqrt(3), 1], [-1, 2]] and B = [[2, sqrt(5)], [1, -3]], then A+B is:
- A. [[sqrt(3)+2, 1+sqrt(5)], [0, -1]]
- B. [[sqrt(3)-2, 1-sqrt(5)], [-2, 5]]
- C. [[2-sqrt(3), sqrt(5)-1], [2, -5]]
- D. [[sqrt(3)+2, 1-sqrt(5)], [0, -1]]
Answer: A. [[sqrt(3)+2, 1+sqrt(5)], [0, -1]] — Matrix addition is performed by adding corresponding elements. So, A+B = [[sqrt(3)+2, 1+sqrt(5)], [-1+1, 2-3]] = [[sqrt(3)+2, 1+sqrt(5)], [0, -1]].
MCQ 5. If A = [[2, 3], [4, 5]], then 3A is:
- A. [[2, 3], [4, 5]]
- B. [[6, 9], [12, 15]]
- C. [[5, 6], [7, 8]]
- D. [[1, 1], [1, 1]]
Answer: B. [[6, 9], [12, 15]] — Scalar multiplication involves multiplying each element of the matrix by the scalar. So, 3A = [[3*2, 3*3], [3*4, 3*5]] = [[6, 9], [12, 15]].
MCQ 6. If [[x+y, 2], [5+z, xy]] = [[6, 2], [5, 8]], then find the values of x, y, and z.
- A. x=4, y=2, z=0
- B. x=2, y=4, z=0
- C. x=3, y=3, z=0
- D. x=6, y=0, z=0
Answer: A. x=4, y=2, z=0 — By equating corresponding elements: x+y=6, 5+z=5, and xy=8. From 5+z=5, we get z=0. For x+y=6 and xy=8, the numbers are 4 and 2 (4+2=6, 4*2=8). Thus, (x,y) can be (4,2) or (2,4). If we choose x=4, y=2, then z=0.
MCQ 7. If A = [[1, 2], [3, 4]] and B = [[1, 0], [0, 1]], then AB is:
- A. A
- B. B
- C. A+B
- D. I (identity matrix)
Answer: A. A — Matrix B is the identity matrix I. The identity matrix acts as the multiplicative identity for matrices, so A * I = A. Therefore, AB = A.
MCQ 8. If A is a matrix of order m x n and B is a matrix of order p x q, then the product AB is defined if:
- A. n = p
- B. m = q
- C. m = p
- D. n = q
Answer: A. n = p — For the product of two matrices AB to be defined, the number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B).
MCQ 9. If A = [[1, 2], [0, 1]], B = [[1, 0], [2, 1]], C = [[2, 0], [1, 2]], then (AB)C is:
- A. [[12, 4], [5, 2]]
- B. [[12, 5], [4, 2]]
- C. [[5, 2], [2, 1]]
- D. [[1, 0], [0, 1]]
Answer: A. [[12, 4], [5, 2]] — First calculate AB = [[1*1+2*2, 1*0+2*1], [0*1+1*2, 0*0+1*1]] = [[5, 2], [2, 1]]. Then, (AB)C = [[5*2+2*1, 5*0+2*2], [2*2+1*1, 2*0+1*2]] = [[12, 4], [5, 2]].
MCQ 10. If A = [[3, sqrt(3), 2], [4, 2, 0]], then A' (transpose of A) is:
- A. [[3, sqrt(3), 2], [4, 2, 0]]
- B. [[3, 4], [sqrt(3), 2], [2, 0]]
- C. [[2, 0], [sqrt(3), 2], [3, 4]]
- D. [[4, 2, 0], [3, sqrt(3), 2]]
Answer: B. [[3, 4], [sqrt(3), 2], [2, 0]] — The transpose of a matrix is obtained by interchanging its rows and columns. So, the first row of A becomes the first column of A', and the second row becomes the second column.
MCQ 11. If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then (A+B)' is:
- A. [[6, 8], [10, 12]]
- B. [[6, 10], [8, 12]]
- C. [[5, 9], [9, 12]]
- D. [[6, 10], [12, 8]]
Answer: B. [[6, 10], [8, 12]] — First, A+B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]. Then, (A+B)' = [[6, 10], [8, 12]]. Alternatively, (A+B)' = A'+B'.
MCQ 12. A square matrix A is called symmetric if:
- A. A = -A'
- B. A = A'
- C. A = A^(-1)
- D. A = I
Answer: B. A = A' — A square matrix A is defined as symmetric if its transpose A' is equal to the matrix A itself (A = A').
MCQ 13. A square matrix A is called skew-symmetric if:
- A. A = A'
- B. A = -A'
- C. A = A^(-1)
- D. A = I
Answer: B. A = -A' — A square matrix A is defined as skew-symmetric if its transpose A' is equal to the negative of the matrix A (A = -A').
MCQ 14. If A = [[2, -2, -4], [-1, 3, 4], [1, -2, -3]], then the symmetric part of A is:
- A. [[2, -3/2, -3/2], [-3/2, 3, 1], [-3/2, 1, -3]]
- B. [[0, -1/2, -5/2], [1/2, 0, 3], [5/2, -3, 0]]
- C. [[2, -1, 1], [-2, 3, -2], [-4, 4, -3]]
- D. [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
Answer: A. [[2, -3/2, -3/2], [-3/2, 3, 1], [-3/2, 1, -3]] — Any square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. The symmetric part is given by (1/2)(A + A'). First, A' = [[2, -1, 1], [-2, 3, -2], [-4, 4, -3]]. Then, A+A' = [[4, -3, -3], [-3, 6, 2], [-3, 2, -6]]. Finally, the symmetric part is (1/2)(A+A') = [[2, -3/2, -3/2], [-3/2, 3, 1], [-3/2, 1, -3]].
MCQ 15. If A and B are matrices of appropriate orders, then (AB)' is equal to:
- A. A'B'
- B. B'A'
- C. AB
- D. BA
Answer: B. B'A' — The transpose of a product of matrices is the product of their transposes in reverse order. This is a fundamental property of matrix transpositions.
MCQ 16. Which of the following is NOT an elementary row operation?
- A. R_i <-> R_j (Swapping two rows)
- B. R_i -> kR_i, where k is a non-zero scalar
- C. R_i -> R_i + kR_j, where k is any scalar
- D. R_i -> 0R_i (Multiplying a row by zero)
Answer: D. R_i -> 0R_i (Multiplying a row by zero) — Elementary row operations allow swapping rows, multiplying a row by a non-zero scalar, or adding a scalar multiple of one row to another. Multiplying a row by zero is not allowed as it can reduce the rank of the matrix, making it non-invertible if it was initially.
MCQ 17. Find the inverse of A = [[2, 1], [1, 1]] using elementary row operations.
- A. [[1, -1], [-1, 2]]
- B. [[-1, 1], [1, -2]]
- C. [[1, 1], [1, 2]]
- D. [[2, 1], [1, 1]]
Answer: A. [[1, -1], [-1, 2]] — Start with the augmented matrix [A|I]. Apply R1 <-> R2, then R2 -> R2 - 2R1, then R2 -> -R2, and finally R1 -> R1 - R2. The resulting matrix on the right side will be A^(-1) = [[1, -1], [-1, 2]].
MCQ 18. A square matrix A is said to be idempotent if:
- A. A^2 = I
- B. A^2 = A
- C. A^2 = 0
- D. A' = A
Answer: B. A^2 = A — An idempotent matrix is a matrix which, when multiplied by itself, yields itself. This is defined by the condition A^2 = A.
MCQ 19. A square matrix A is said to be nilpotent of index k if:
- A. A^k = I
- B. A^k = A
- C. A^k = 0 and A^(k-1) != 0
- D. A' = A
Answer: C. A^k = 0 and A^(k-1) != 0 — A nilpotent matrix of index k is a matrix A such that A^k is the null matrix (0) and A^(k-1) is not the null matrix. The smallest such k is the index.
MCQ 20. A square matrix A is said to be involutory if:
- A. A^2 = I
- B. A^2 = A
- C. A^2 = 0
- D. A' = A
Answer: A. A^2 = I — An involutory matrix is a matrix that is its own inverse, meaning that when multiplied by itself, it results in the identity matrix (A^2 = I).
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.