Matrices

CBSE Class 12 Maths · 11 questions · 26 marks

A matrix packages a rectangular array of numbers into a single object that can be added, scaled and multiplied under its own rules. This chapter fixes the arithmetic — especially the non-commutative product AB ≠ BA in general — and introduces the transpose, which splits every square matrix into a symmetric and a skew-symmetric part. Elementary row operations then give a mechanical way to invert a matrix without touching determinants.

Quick revision

A matrix of order m×n has m rows and n columns; addition and subtraction require matrices of the same order.
Matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B; in general AB ≠ BA.
Transpose A' is obtained by interchanging rows and columns; (A')' = A, (A+B)' = A'+B', and (AB)' = B'A'.
A is symmetric if A' = A; A is skew-symmetric if A' = −A. Every diagonal entry of a skew-symmetric matrix must be 0.
Every square matrix A can be written uniquely as A = P + Q, where P = (A+A')/2 is symmetric and Q = (A−A')/2 is skew-symmetric.
The identity matrix I satisfies AI = IA = A; a scalar matrix is a diagonal matrix with all diagonal entries equal.
A square matrix A is invertible if there exists B with AB = BA = I; the inverse, when it exists, is unique and written A⁻¹.
Elementary row operations (Rᵢ↔Rⱼ, Rᵢ→kRᵢ, Rᵢ→Rᵢ+kRⱼ) applied to [A | I] until the left block becomes I turn the right block into A⁻¹.
If A is an m×n matrix and B is n×p, the product AB has order m×p.

Important questions

1
MCQ1 mark
If A is a matrix of order 2×3 and B is a matrix of order 3×4, then the order of AB is:
1. A.2×4
2. B.3×3
3. C.4×2
4. D.2×3
Answer
The order of AB is 2×4. Matrix multiplication is defined when the number of columns of A (3) equals the number of rows of B (3), which holds here. The resulting product takes the number of rows of A and the number of columns of B, giving order 2×4.
2
MCQ1 mark
If A = [[2,3],[1,4]], then A' (the transpose of A) is:
1. A.[[2,1],[3,4]]
2. B.[[2,3],[1,4]]
3. C.[[4,1],[3,2]]
4. D.[[1,2],[4,3]]
Answer
A' = [[2,1],[3,4]]. The transpose is formed by turning rows into columns: the first row (2,3) becomes the first column, and the second row (1,4) becomes the second column. So A' = [[2,1],[3,4]].
3
MCQ1 mark
For a skew-symmetric matrix A, the diagonal elements are always:
1. A.1
2. B.0
3. C.equal to the off-diagonal elements
4. D.undefined
Answer
The diagonal elements are always 0. A skew-symmetric matrix satisfies A' = −A, so every entry aᵢⱼ = −aⱼᵢ. For a diagonal entry, i = j, giving aᵢᵢ = −aᵢᵢ, so 2aᵢᵢ = 0 and aᵢᵢ = 0.
4
MCQ1 mark
If A = [[1,2],[3,4]] and B = [[2,0],[1,2]], then AB equals:
1. A.[[4,4],[10,8]]
2. B.[[2,4],[3,8]]
3. C.[[4,0],[10,8]]
4. D.[[4,4],[6,8]]
Answer
AB = [[4,4],[10,8]]. Row 1 of AB: (1×2 + 2×1, 1×0 + 2×2) = (4, 4). Row 2 of AB: (3×2 + 4×1, 3×0 + 4×2) = (10, 8). So AB = [[4,4],[10,8]].
5
Assertion & Reason1 mark
Assertion (A): For any square matrix A, A + A' is always symmetric. Reason (R): (A+A')' = A' + (A')' = A' + A.
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 correctly expands (A+A')' using (X+Y)' = X'+Y' and (A')' = A, giving A' + A. Since A' + A = A + A', this shows (A+A')' = A+A', which is exactly the condition for A+A' to be symmetric — proving Assertion A and matching it directly.
6
Short answer2 marks
Find x and y if [[x+y, 2], [5, y]] = [[6, 2], [5, 3]].
7
Short answer2 marks
If A = [[3,-2],[4,1]], verify that A + A' is a symmetric matrix.
8
Short answer3 marks
If A = [[1,2],[2,1]] and I is the 2×2 identity matrix, show that A² − 2A − 3I = O (the zero matrix).
9
Long answer5 marks
Express the matrix A = [[3,5],[1,-1]] as the sum of a symmetric matrix and a skew-symmetric matrix.
10
Long answer5 marks
Using elementary row operations, find the inverse of A = [[2,1],[1,1]].
11
Case study4 marks
Read the following and answer the questions that follow: A stationery store sells notebooks and pens from two outlets. In week 1, matrix A = [[40,25],[30,20]] gives the units sold, where rows represent Store P and Store Q and columns represent notebooks and pens. In week 2, matrix B = [[35,30],[25,15]] gives the same information. (a) Find the total units sold over both weeks, A + B. (b) Each notebook costs ₹20 and each pen costs ₹10. Write the price column matrix C. (c) Using matrix multiplication, find the week-1 revenue for each store, A×C. (d) State the order of the revenue matrix found in (c).

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Matrices

CBSE Class 12 Maths · 11 questions · 26 marks

Quick revision

A matrix of order m×n has m rows and n columns; addition and subtraction require matrices of the same order.

Matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B; in general AB ≠ BA.

Transpose A' is obtained by interchanging rows and columns; (A')' = A, (A+B)' = A'+B', and (AB)' = B'A'.

A is symmetric if A' = A; A is skew-symmetric if A' = −A. Every diagonal entry of a skew-symmetric matrix must be 0.

Every square matrix A can be written uniquely as A = P + Q, where P = (A+A')/2 is symmetric and Q = (A−A')/2 is skew-symmetric.

The identity matrix I satisfies AI = IA = A; a scalar matrix is a diagonal matrix with all diagonal entries equal.

A square matrix A is invertible if there exists B with AB = BA = I; the inverse, when it exists, is unique and written A⁻¹.

Elementary row operations (Rᵢ↔Rⱼ, Rᵢ→kRᵢ, Rᵢ→Rᵢ+kRⱼ) applied to [A | I] until the left block becomes I turn the right block into A⁻¹.

If A is an m×n matrix and B is n×p, the product AB has order m×p.

Important questions

MCQ1 mark

If A is a matrix of order 2×3 and B is a matrix of order 3×4, then the order of AB is:

A.2×4
B.3×3
C.4×2
D.2×3

MCQ1 mark

If A = [[2,3],[1,4]], then A' (the transpose of A) is:

A.[[2,1],[3,4]]
B.[[2,3],[1,4]]
C.[[4,1],[3,2]]
D.[[1,2],[4,3]]

MCQ1 mark

For a skew-symmetric matrix A, the diagonal elements are always:

A.1
B.0
C.equal to the off-diagonal elements
D.undefined

MCQ1 mark

If A = [[1,2],[3,4]] and B = [[2,0],[1,2]], then AB equals:

A.[[4,4],[10,8]]
B.[[2,4],[3,8]]
C.[[4,0],[10,8]]
D.[[4,4],[6,8]]

Assertion & Reason1 mark

Assertion (A): For any square matrix A, A + A' is always symmetric. Reason (R): (A+A')' = A' + (A')' = A' + A.

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

Find x and y if [[x+y, 2], [5, y]] = [[6, 2], [5, 3]].

Short answer2 marks

If A = [[3,-2],[4,1]], verify that A + A' is a symmetric matrix.

Short answer3 marks

If A = [[1,2],[2,1]] and I is the 2×2 identity matrix, show that A² − 2A − 3I = O (the zero matrix).

Long answer5 marks

Express the matrix A = [[3,5],[1,-1]] as the sum of a symmetric matrix and a skew-symmetric matrix.

Long answer5 marks

Using elementary row operations, find the inverse of A = [[2,1],[1,1]].

Case study4 marks

Read the following and answer the questions that follow: A stationery store sells notebooks and pens from two outlets. In week 1, matrix A = [[40,25],[30,20]] gives the units sold, where rows represent Store P and Store Q and columns represent notebooks and pens. In week 2, matrix B = [[35,30],[25,15]] gives the same information. (a) Find the total units sold over both weeks, A + B. (b) Each notebook costs ₹20 and each pen costs ₹10. Write the price column matrix C. (c) Using matrix multiplication, find the week-1 revenue for each store, A×C. (d) State the order of the revenue matrix found in (c).