Class 12 · Chapter 3 · Algebra unit (10 of 80 marks)

Matrices Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 63 board-style questions.

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63 questions
20 multiple choice, 4 assertion–reason, 17 very short answer, 13 short answer, 5 long answer, 4 case study
About 7 hours to master

Unit II Algebra (Matrices + Determinants): 10 marks of the 80-mark paper — CBSE Curriculum 2025-26 Mathematics (041) (official PDF).

NCERT textbook: Chapter 3 on ncert.nic.in (PDF)

Revision notes

Matrices — revision notes

1. Order, construction and types

An $m \times n$ matrix has $m$ rows and $n$ columns; $a_{i j}$ sits in row $i$ , column $j$ . A matrix with $k$ elements can have order $m \times n$ for every factor pair $m n = k$ .
Types: row, column, square, diagonal ( $a_{i j} = 0$ for $i \neq j$ ), scalar (diagonal with equal diagonal entries, $k I$ ), identity $I$ , zero matrix $O$ .
Equal matrices: same order and equal corresponding entries — equate entries to get equations.

2. Operations

$A + B$ and $k A$ : entry by entry, same order only. $A + B = B + A$ , $k (A + B) = k A + k B$ .
$A B$ is defined when (columns of $A$ ) = (rows of $B$ ); $(m \times n) (n \times p) = m \times p$ . Entry $(i, j)$ = row $i$ of $A$ · column $j$ of $B$ .
Associative and distributive, but not commutative: in general $A B \neq B A$ , so $(A + B)^{2} = A^{2} + A B + B A + B^{2}$ .
$A B = O$ does not force $A = O$ or $B = O$ (e.g. $[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}] = O$ ).

3. Transpose, symmetric and skew-symmetric matrices

$(A^{T})^{T} = A$ , $(A + B)^{T} = A^{T} + B^{T}$ , $(k A)^{T} = k A^{T}$ , $(A B)^{T} = B^{T} A^{T}$ (order reverses).
Symmetric: $A^{T} = A$ . Skew-symmetric: $A^{T} = - A$ (diagonal entries all $0$ ).
Every square matrix: $A = \frac{1}{2} (A + A^{T}) + \frac{1}{2} (A - A^{T})$ = symmetric + skew-symmetric, uniquely.
To test a product, take its transpose and use $(X Y)^{T} = Y^{T} X^{T}$ with $A^{T} = \pm A$ .

4. Invertible matrices

$B$ is the inverse of square $A$ if $A B = B A = I$ . The inverse is unique: $B = B (A C) = (B A) C = C$ .
$(A B)^{- 1} = B^{- 1} A^{- 1}$ . If $A$ satisfies $A^{2} + p A + q I = O$ with $q \neq 0$ , then $A (- \frac{1}{q} (A + p I)) = I$ .
If $A$ is invertible and $A X = C$ , then $X = A^{- 1} C$ .

Worked example 1

$A = [\begin{matrix} 1 & 3 \\ 0 & 1 \end{matrix}]$ : $A^{2} = [\begin{matrix} 1 & 6 \\ 0 & 1 \end{matrix}]$ , and in general $A^{n} = [\begin{matrix} 1 & 3 n \\ 0 & 1 \end{matrix}]$ (each multiplication adds $3$ to the corner).

Worked example 2

If $A$ is symmetric, $B^{T} A B$ is symmetric: $(B^{T} A B)^{T} = B^{T} A^{T} (B^{T})^{T} = B^{T} A B$ .

Worked example 3

$A = [\begin{matrix} 4 & - 1 \\ 2 & 1 \end{matrix}]$ satisfies $A^{2} - 5 A + 6 I = O$ (check: $A^{2} = [\begin{matrix} 14 & - 5 \\ 10 & - 1 \end{matrix}]$ ). So $A (5 I - A) = 6 I$ and $A^{- 1} = \frac{1}{6} [\begin{matrix} 1 & 1 \\ - 2 & 4 \end{matrix}]$ .

Common errors

Writing $(A B)^{T} = A^{T} B^{T}$ — the order must reverse.
Expanding $(A + B)^{2}$ as $A^{2} + 2 A B + B^{2}$ or cancelling: $A B = A C$ does not give $B = C$ unless $A$ is invertible.
Multiplying matrices whose orders do not match, or mixing up rows × columns when stating an order.
Checking only $A B = I$ when the question asks you to verify an inverse: show $B A = I$ as well.
Forgetting that a skew-symmetric matrix has zero diagonal.

Board-exam tips

Write the order of each product before multiplying — it catches most errors.
In “show that” questions, show the full matrix after each multiplication; step marks are given for $A^{2}$ and for the final combination separately.
Elementary row operations are not in the current syllabus; use the definition $A B = B A = I$ or a polynomial identity.

Topics in this chapter: Matrix, order and construction · Types of matrices and equality · Matrix multiplication and its properties · Transpose of a matrix · Symmetric and skew-symmetric matrices · Invertible matrices · Addition and scalar multiplication.

Formulas for this chapter: Chapter 3 on the Class 12 formula sheet (free PDF and A3 poster).

Matrices Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 173 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceOrder and types of matrices

$A$ has order $3 \times 2$ and $B$ has order $2 \times 4$ . Which statement is true?

Show worked answer

$(3 \times 2) (2 \times 4)$ : inner orders match, so $A B$ is $3 \times 4$ ; $(2 \times 4) (3 \times 2)$ : $4 \neq 3$ , so $B A$ does not exist A1
Correct option: (a) $A B$ exists and has order $3 \times 4$ .

Q2·1 mark·Multiple choiceTranspose, symmetric and skew-symmetric matrices

The matrix $[\begin{matrix} 2 & x \\ 3 & 4 \end{matrix}]$ is symmetric. Then $x$ equals

Q3·1 mark·Assertion–reasonTranspose, symmetric and skew-symmetric matrices

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): For every square matrix $A$ , the matrix $A + A^{'}$ is symmetric.

Reason (R): For matrices of the same order, $(A^{'})^{'} = A$ and $(A + B)^{'} = A^{'} + B^{'}$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerOperations on matrices

Show that $A = [\begin{matrix} 2 & 1 \\ - 1 & 3 \end{matrix}]$ satisfies $A^{2} - 5 A = k I$ for some number $k$ , and find $k$ .

Show worked answer

Answer:

k = - 7

$A^{2} = [\begin{matrix} 3 & 5 \\ - 5 & 8 \end{matrix}]$ and $5 A = [\begin{matrix} 10 & 5 \\ - 5 & 15 \end{matrix}]$ M1
$\begin{aligned} A^{2} - 5 A & = [\begin{array}{c} - 7 & 0 \\ 0 & - 7 \end{array}] \\ = - 7 I \end{aligned}$ , so $k = - 7$ A1

Q5·2 marks·Short answerOrder and types of matrices

Find $x$ and $y$ if $[\begin{matrix} x + y & 2 \\ 5 & x - y \end{matrix}] = [\begin{matrix} 6 & 2 \\ 5 & 2 \end{matrix}]$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerTranspose, symmetric and skew-symmetric matrices

Write $A = [\begin{matrix} 6 & - 2 \\ 8 & 1 \end{matrix}]$ as $P + Q$ , where $P$ is symmetric and $Q$ is skew-symmetric.

Show worked answer

Answer:

[\begin{matrix} 6 & 3 \\ 3 & 1 \end{matrix}] + [\begin{matrix} 0 & - 5 \\ 5 & 0 \end{matrix}]

$\begin{aligned} P & = \frac{1}{2} (A + A^{'}) \\ = \frac{1}{2} [\begin{array}{c} 12 & 6 \\ 6 & 2 \end{array}] \\ = [\begin{array}{c} 6 & 3 \\ 3 & 1 \end{array}] \end{aligned}$ , and $P^{'} = P$ A1
$\begin{aligned} Q & = \frac{1}{2} (A - A^{'}) \\ = \frac{1}{2} [\begin{array}{c} 0 & - 10 \\ 10 & 0 \end{array}] \\ = [\begin{array}{c} 0 & - 5 \\ 5 & 0 \end{array}] \end{aligned}$ , and $Q^{'} = - Q$ A1
$\begin{aligned} P + Q & = [\begin{array}{c} 6 & - 2 \\ 8 & 1 \end{array}] \\ = A \end{aligned}$ A1

Q7·3 marks·Short answerTranspose, symmetric and skew-symmetric matrices

For $A = [\begin{matrix} 2 & - 1 \\ 3 & 1 \end{matrix}]$ and $B = [\begin{matrix} 1 & 4 \\ 0 & - 2 \end{matrix}]$ , verify that $(A B)^{'} = B^{'} A^{'}$ .

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerInvertible matrices

Let $A = [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}]$ . (a) Show that $A^{2} - 5 A + 7 I = O$ . (b) Use this to find $A^{- 1}$ .

Show worked answer

Answer:

A^{- 1} = \frac{1}{7} [\begin{matrix} 2 & - 1 \\ 1 & 3 \end{matrix}]

(a) $A^{2} = [\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}]$ A1
$\begin{aligned} - 5 A + 7 I & = [\begin{array}{c} - 15 + 7 & - 5 \\ 5 & - 10 + 7 \end{array}] \\ = [\begin{array}{c} - 8 & - 5 \\ 5 & - 3 \end{array}] \end{aligned}$ , so $A^{2} - 5 A + 7 I = O$ A1
(b) Multiply by $A^{- 1}$ : $A - 5 I + 7 A^{- 1} = O$ M1
$A^{- 1} = \frac{1}{7} (5 I - A)$ A1
$= \frac{1}{7} [\begin{matrix} 2 & - 1 \\ 1 & 3 \end{matrix}]$ A1

Q9·5 marks·Long answerOperations on matrices

Shop P sold $10$ pens, $5$ notebooks and $2$ bags in a day; shop Q sold $8$ pens, $6$ notebooks and $3$ bags. The selling prices are ₹ $10$ a pen, ₹ $40$ a notebook and ₹ $200$ a bag, and the cost prices are ₹ $8$ , ₹ $30$ and ₹ $150$ . Using matrices, find each shop's takings and each shop's profit for the day.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyOperations on matrices

Canteen orders. A school canteen records orders in $O = [\begin{matrix} 2 & 1 \\ 1 & 3 \end{matrix}]$ (rows: day 1, day 2; columns: sandwiches, juices). A sandwich costs ₹ $40$ and a juice ₹ $25$ , written as $P = [\begin{matrix} 40 \\ 25 \end{matrix}]$ .

Write the order of the matrix $O P$ . [1 mark]
Find the bill for day 1. [1 mark]
Find the total bill for both days, and say whether $O$ is symmetric. [2 marks]

Show worked answer

$(2 \times 2) (2 \times 1)$ gives $2 \times 1$ A1
$2 \times 40 + 1 \times 25 =$ ₹ $105$ A1
Day 2: $40 + 75 =$ ₹ $115$ ; total ₹ $220$ M1
$O^{'} = O$ , so $O$ is symmetric A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

Read and construct matrices, state orders, use equality, add, scale and multiply matrices correctly.

Read first: 1. Order, construction and types; 2. Operations 6 practice questions · checkpoint: 3 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

Handle transposes, symmetric/skew-symmetric decomposition, matrix powers and inverses from the definition at board 2- and 3-mark level.

Read first: 3. Transpose, symmetric and skew-symmetric matrices; 4. Invertible matrices 23 practice questions · checkpoint: 4 questions, 8 marks, pass 75%

Practise step 2

Step 3

Full marks on long answers

Write complete, step-marked solutions to 5-mark and case-based matrix questions, including context problems.

Read first: Worked examples 1-3; Board-exam tips 5 practice questions · checkpoint: 3 questions, 15 marks, pass 70%

Practise step 3

Step 4

95+ stretch (HOTS)

Solve unfamiliar matrix problems: matrix equations with unknown entries, nilpotent/idempotent reasoning and pattern arguments for powers.

Read first: 2. Operations (non-commutativity); 4. Invertible matrices 6 practice questions · checkpoint: 3 questions, 13 marks, pass 60%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 7 of the 63 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceTypes of matrices and equality

The $3 \times 3$ matrix $A = [a_{i j}]$ with $a_{i j} = i^{2} - j^{2}$ is

(a)a skew-symmetric matrix
(b)a symmetric matrix
(c)a diagonal matrix
(d)a scalar matrix

Q2·1 mark·Multiple choiceMatrix multiplication and its properties

$A$ , $B$ , $C$ are matrices of orders $2 \times 3$ , $3 \times 2$ and $2 \times 2$ respectively. Which of the following is defined?

(a) $A C$
(b) $B A + C$
(c) $A B + C$
(d) $C B$

Q3·1 mark·Multiple choiceTypes of matrices and equality

If $[\begin{matrix} 2 x - y & 3 \\ x + y & z \end{matrix}] = [\begin{matrix} 5 & 3 \\ 7 & 2 z - 4 \end{matrix}]$ , then $x + y + z$ equals

(a) $9$
(b) $11$
(c) $15$
(d) $7$

Worked solution·2 marks·Very short answerSymmetric and skew-symmetric matrices

The symmetric part $\frac{1}{2} (A + A^{T})$ of a $2 \times 2$ matrix $A$ is $[\begin{matrix} 2 & 1 \\ 1 & 0 \end{matrix}]$ , and the element $a_{12}$ of $A$ is $4$ . Find $A$ .

Full worked solution Free, no sign-in

$A = P + Q$ with $P = [\begin{matrix} 2 & 1 \\ 1 & 0 \end{matrix}]$ and $Q = [\begin{matrix} 0 & k \\ - k & 0 \end{matrix}]$ skew-symmetric; $a_{12} = 1 + k = 4 \Rightarrow k = 3$ M1
$A = [\begin{matrix} 2 & 4 \\ - 2 & 0 \end{matrix}]$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S24 · Problem S88 · Problem S23 · Problem S87. All 22 →

Knowledge organiser: Matrices on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Matrices

Writing (AB)ᵀ = AᵀBᵀ. Fix: always reverse the order, (AB)ᵀ = BᵀAᵀ, and check the orders of the factors.
Expanding (A + B)² as A² + 2AB + B². Fix: write A² + AB + BA + B² and only combine if AB = BA is given.
Arithmetic slips in 3×3 products that wreck the rest of a 5-mark answer. Fix: write each entry as a row·column sum and re-check one row by a second method (e.g. trace or a column sum).
Stating the order as columns × rows. Fix: order is always rows × columns; write the order of every product before multiplying.
Verifying an inverse with only AB = I. Fix: show both AB = I and BA = I (one line each).
Decomposition answers where the "skew" part has non-zero diagonal. Fix: check P = Pᵀ, Qᵀ = −Q and P + Q = A before moving on.
Using elementary row operations (deleted) and losing time. Fix: use the definition or a polynomial identity A² + pA + qI = O.

Exam strategy: how answers are step-marked · the proofs, written out · competency-based questions

Matrices in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 5, 6
Sample paper 2 (Board standard): questions 3, 5
Sample paper 3 (Above board): questions 3, 5, 6
Sample paper 4 (Above board): questions 3, 6
Sample paper 5 (95+ challenge): questions 3, 5, 32

Want it against the clock? Take a timed 30-mark chapter test on Matrices, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on matrices and determinants.

Matrices: common questions

What is in CBSE Class 12 Maths Chapter 3, Matrices?

The topics are: Matrix, order and construction, Types of matrices and equality, Matrix multiplication and its properties, Transpose of a matrix, Symmetric and skew-symmetric matrices, Invertible matrices, Addition and scalar multiplication. This page has revision notes for each, and 63 board-style practice questions with step-marked solutions.

How many marks is Matrices worth?

It is in the Algebra unit, which carries 10 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Matrices?

Writing (AB)ᵀ = AᵀBᵀ. Fix: always reverse the order, (AB)ᵀ = BᵀAᵀ, and check the orders of the factors. Expanding (A + B)² as A² + 2AB + B². Fix: write A² + AB + BA + B² and only combine if AB = BA is given.

Are the Matrices important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Matrices — revision notes

1. Order, construction and types

2. Operations

3. Transpose, symmetric and skew-symmetric matrices

4. Invertible matrices

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Matrices Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Matrices

Matrices in our sample papers

Matrices: common questions

Work through the Route to 95 with your progress saved