MathsCBSE Class 12 · Matrices

Multiplication of Matrices: Rules, Properties and Solved Questions (Class 12)

Updated 15 October 2026 · Fact-checked

The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, AB is m × p; its (i, j) entry is row i of A times column j of B, term by term, added. In general AB ≠ BA.

Check that the inner numbers match, then walk along a row of A and down a column of B, multiply pairs and add to get one entry.

Illustration drawn for QuestionBuddy.

Understand Multiplication of Matrices

Matrix multiplication is the operation that makes matrices useful: it combines quantities with rates, prices with amounts, or one step of a process with the next. It is also the operation with the most traps, so learn it carefully.

When is AB defined? Write the orders side by side: $A_{m \times n} B_{n \times p}$ . The two inner numbers must match (columns of A = rows of B); the outer numbers give the order of the product, $m \times p$ . So a $2 \times 3$ matrix times a $3 \times 2$ matrix gives a $2 \times 2$ matrix, while the reverse product $B A$ is $3 \times 3$ . If the inner numbers do not match, the product does not exist.

How to compute it. The entry in row $i$ and column $j$ of $A B$ is
$c_{ij} = a_{i 1} b_{1 j} + a_{i 2} b_{2 j} + \dots + a_{in} b_{nj}$ ,
that is, walk along row $i$ of A and down column $j$ of B, multiply matching entries and add. Place your left finger on the row and your right finger on the column. Work entry by entry and write each sum before simplifying; most lost marks are arithmetic slips with negative signs.

Properties that hold (when the products are defined):

Associative: $(A B) C = A (B C)$ .
Distributive: $A (B + C) = A B + A C$ and $(A + B) C = A C + B C$ .
Identity: $A I = I A = A$ for a square matrix A and the identity I of the same order.
Scalars move freely: $(k A) B = A (k B) = k (A B)$ .

Properties that fail.

Not commutative: in general $A B \neq = B A$ , even for square matrices of the same order. Sometimes only one product exists, sometimes both exist with different orders, and even when both are $2 \times 2$ they usually differ. So $(A + B)^{2} = A^{2} + A B + B A + B^{2}$ , which equals $A^{2} + 2 A B + B^{2}$ only when $A B = B A$ ; likewise $(A + B) (A - B) = A^{2} - A B + B A - B^{2}$ .
Zero products: $A B = O$ does not force $A = O$ or $B = O$ . For example $[1111] [1 - 1 - 1 1] = O$ . Also, $A B = A C$ does not let you cancel A.

The 2026-27 syllabus restricts these two 'failure' facts to square matrices of order 2, so your examples should be $2 \times 2$ .

Powers and matrix polynomials. For a square matrix, $A^{2} = AA$ and $A^{3} = A^{2} A$ . A matrix polynomial such as $A^{2} - 5 A + 7 I$ is evaluated by replacing the constant term with a multiple of $I$ . A popular board question asks you to show $A^{2} - k A + m I = O$ or to find $k$ ; once proved, such a relation lets you write $A^{3}$ , or later $A^{- 1}$ , in terms of $A$ and $I$ without further multiplication.

Matrix multiplication is NCERT Section 3.4.5 (Chapter 3) and feeds directly into determinants, inverses and solving equations by the matrix method.

Multiplication of Matrices: formulas to remember

Condition and order: $A_{m \times n} B_{n \times p} = (A B)_{m \times p}$; Inner numbers equal; outer numbers give the order.
Entry of the product: $(A B)_{ij} = \sum_{k = 1}^{n} a_{ik} b_{k j}$; Row i of A times column j of B.
Associative and distributive laws: $(A B) C = A (B C), A (B + C) = A B + A C$; Valid whenever the products are defined.
Identity: $A I = I A = A$; I has the same order as the square matrix A.
Square of a sum: $(A + B)^{2} = A^{2} + A B + B A + B^{2}$; Equals $A^{2} + 2 A B + B^{2}$ only if $A B = B A$ .
Non-commutativity: $A B \neq = B A in general$; Check by computing both, or by comparing orders.
Matrix polynomial: $f (x) = x^{2} - k x + m \Rightarrow f (A) = A^{2} - k A + m I$; The constant becomes a multiple of the identity.

How to solve Multiplication of Matrices questions

Use this for computing products, finding unknowns from a product, and matrix polynomial questions.

1Write the orders of the matrices and check that the inner numbers match; write the order of the product.
2For each entry, write the full sum of products (for example $1 \cdot 2 + 2 \cdot (- 1) + (- 1) \cdot 3$ ) before simplifying.
3If both $A B$ and $B A$ are asked, compute them separately; never assume they are equal.
4For a polynomial like $A^{2} - 6 A + 5 I$ , compute $A^{2}$ first, then $6 A$ and $5 I$ , and subtract entry by entry.
5To find unknowns from a product such as $[x 21] M [x 1 - 2]^{T} = O$ , multiply from left to right, keeping the order of the matrices, and equate the result to zero.
6To find a constant $k$ in $A^{2} = k A + m I$ , compare one convenient entry (usually an off-diagonal one) and check the others.
7When you prove $A^{2} = k A + m I$ , use it to get higher powers: multiply both sides by A and substitute again.

Multiplication of Matrices in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Product of a quantity row and a price matrix

Suppose a family buys 2 kg of apples, 3 kg of bananas and 1 kg of grapes every week, and two nearby shops X and Y charge (per kg) Rs 120 and Rs 110 for apples, Rs 40 and Rs 45 for bananas, and Rs 60 and Rs 50 for grapes.

Write the weekly quantities as a $1 \times 3$ matrix and the prices as a $3 \times 2$ matrix (one column per shop). Use matrix multiplication to find the weekly bill at each shop and say which is cheaper.

Try it first, then show the solutionSolution

$Q = [231]$ and $P = 12040601104550$ . Orders $1 \times 3$ and $3 \times 2$ , so $QP$ is $1 \times 2$ .
Shop X: $2 (120) + 3 (40) + 1 (60) = 240 + 120 + 60 = 420$ .
Shop Y: $2 (110) + 3 (45) + 1 (50) = 220 + 135 + 50 = 405$ .
$QP = [420405]$ .

Answer: Rs 420 at shop X and Rs 405 at shop Y, so shop Y is cheaper by Rs 15 a week.

Memory hook: Quantities times prices, row into column: that is a shopping bill.

Repeated multiplication and powers

Imagine two towns, A and B. Each year 10% of A's residents move to B and 20% of B's residents move to A; nobody else moves. If $a$ and $b$ are this year's populations, next year's are given by $[0.9 0.1 0.2 0.8] [a b]$ .

Starting with 1200 people in A and 300 in B, find the populations after one year and after two years. Check your two-year answer using $M^{2}$ .

Try it first, then show the solutionSolution

Year 1: $0.9 (1200) + 0.2 (300) = 1080 + 60 = 1140$ ; $0.1 (1200) + 0.8 (300) = 120 + 240 = 360$ .
Year 2: $0.9 (1140) + 0.2 (360) = 1026 + 72 = 1098$ ; $0.1 (1140) + 0.8 (360) = 114 + 288 = 402$ .
$M^{2} = [0.81 + 0.02 0.09 + 0.08 0.18 + 0.16 0.02 + 0.64] = [0.83 0.17 0.34 0.66]$ .
$M^{2} [1200300] = [996 + 102 204 + 198] = [1098402]$ , matching.

Answer: After one year: 1140 and 360. After two years: 1098 and 402 (the total stays 1500).

Memory hook: Two years of moving = multiply by M twice = multiply once by $M^{2}$ ; that is associativity at work.

Tricks to remember Multiplication of Matrices

Short memory aids. Say each one aloud once, then use it on a question the same day.

Inner match, outer order
Write $(m \times n) (n \times p)$ : if the inner pair matches, the product exists, and the outer pair is its order.
Use it when: Order and conformability MCQs.
Row runs, column falls
Your left finger runs along a row of the first matrix; your right finger falls down a column of the second. Multiply what they touch, then add.
Use it when: Computing every entry without mixing rows and columns.
Order matters: AB is not BA
Putting on socks then shoes is not shoes then socks. Matrix products are actions in order.
Use it when: Expanding $(A + B)^{2}$ and $(A + B) (A - B)$ correctly.
Constants wear an I
In a matrix polynomial, a plain number must wear an identity matrix: $7$ becomes $7 I$ .
Use it when: Questions like 'show that $A^{2} - 5 A + 7 I = O$ '.

Multiplication of Matrices: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Product of a 2 × 3 and a 3 × 2 matrix · 2 marks · Very short answerBoard level

Let $A = [1023 - 1 2]$ and $B = 2 - 1 3 104$ . Find $A B$ . Are $A B$ and $B A$ of the same order?

Try it first, then show the solutionSolution

Orders: $A$ is $2 \times 3$ and $B$ is $3 \times 2$ , so $A B$ is $2 \times 2$ and $B A$ is $3 \times 3$ . (½ mark)
Row 1: $(1) (2) + (2) (- 1) + (- 1) (3) = - 3$ ; $(1) (1) + (2) (0) + (- 1) (4) = - 3$ . (½ mark)
Row 2: $(0) (2) + (3) (- 1) + (2) (3) = 3$ ; $(0) (1) + (3) (0) + (2) (4) = 8$ . (½ mark)
$A B = [- 3 3 - 3 8]$ ; $A B$ and $B A$ have different orders, so they cannot be equal. (½ mark)

Answer: $A B = [- 3 3 - 3 8]$ ; $A B$ is $2 \times 2$ while $B A$ is $3 \times 3$ .

Example 2: Find k in a matrix polynomial · 3 marks · Short answerBoard level

If $A = [2134]$ , find $k$ such that $A^{2} = k A - 5 I$ .

Try it first, then show the solutionSolution

$A^{2} = [4 + 3 2 + 4 6 + 12 3 + 16] = [761819]$ . (1 mark)
$k A - 5 I = [2 k - 5 k 3 k 4 k - 5]$ . (½ mark)
Equating the (1, 2) entries: $3 k = 18$ , so $k = 6$ . (½ mark)
Check the rest: $2 (6) - 5 = 7$ , $k = 6$ , $4 (6) - 5 = 19$ . All entries match. (1 mark)

Answer: $k = 6$ , that is $A^{2} = 6 A - 5 I$ .

Example 3: Two non-zero matrices with a zero product · 3 marks · Short answerHarder board level

(i) Show that $P = [1021]$ and $Q = [1301]$ do not commute. (ii) Give two non-zero $2 \times 2$ matrices whose product is the zero matrix.

Try it first, then show the solutionSolution

(i) $P Q = [1 + 6 0 + 3 0 + 2 0 + 1] = [7321]$ . (½ mark)
$QP = [1 + 0 3 + 0 2 + 0 6 + 1] = [1327]$ . (½ mark)
The (1, 1) entries differ, so $P Q \neq = QP$ . (½ mark)
(ii) Take $A = [1111]$ and $B = [1 - 1 - 1 1]$ , both non-zero. (½ mark)
$A B = [1 - 1 1 - 1 - 1 + 1 - 1 + 1] = [0000]$ . (1 mark)

Answer: (i) $P Q = [7321] \neq = QP = [1327]$ . (ii) $A = [1111]$ , $B = [1 - 1 - 1 1]$ give $A B = O$ .

Example 4: An unknown inside a triple product · 3 marks · Short answerHarder board level

Find $x$ if $[x 21] 102021210 x 1 - 2 = O$ .

Try it first, then show the solutionSolution

First product ( $1 \times 3$ times $3 \times 3$ ): $[x + 0 + 2, 0 + 4 + 1, 2 x + 2 + 0] = [x + 2, 5, 2 x + 2]$ . (1 mark)
Second product ( $1 \times 3$ times $3 \times 1$ ): $(x + 2) x + 5 (1) + (2 x + 2) (- 2) = x^{2} + 2 x + 5 - 4 x - 4$ . (1 mark)
$= x^{2} - 2 x + 1 = (x - 1)^{2} = 0$ , so $x = 1$ . (1 mark)

Answer: $x = 1$

Example 5: When does (A + B)² equal A² + B²? · 5 marks · Long answerHOTS / competency

Let $A = [1 - 1 2 - 1]$ and $B = [a b 1 - 1]$ . Find $a$ and $b$ such that $(A + B)^{2} = A^{2} + B^{2}$ .

Try it first, then show the solutionSolution

$(A + B)^{2} = A^{2} + A B + B A + B^{2}$ , so the condition is $A B + B A = O$ . (1 mark)
$A B = [a + 2 b - a - b 1 - 2 - 1 + 1] = [a + 2 b - a - b - 1 0]$ . (1 mark)
$B A = [a - 1 b + 1 2 a - 1 2 b + 1]$ . (1 mark)
$A B + B A = [2 a + 2 b - 1 - a + 1 2 a - 2 2 b + 1] = O$ gives $2 a - 2 = 0$ , $- a + 1 = 0$ and $2 b + 1 = 0$ : $a = 1$ , $b = - \frac{1}{2}$ . (1 mark)
Check the (1, 1) entry: $2 (1) + 2 (- \frac{1}{2}) - 1 = 0$ . All four entries vanish. (1 mark)

Answer: $a = 1$ , $b = - \frac{1}{2}$

Example 6: Case study: a school canteen's sales · 4 marks · Case-basedHOTS / competency

Suppose a school canteen records sales of samosas, juice packs and sandwiches over three days. Day 1: 40, 30, 20; Day 2: 35, 25, 30; Day 3: 50, 20, 25. The prices are Rs 15, Rs 20 and Rs 30 respectively.
(i) Write the sales as a matrix S and the prices as a column matrix P, and state the order of SP. (1 mark)
(ii) Find the revenue on each day using SP. (2 marks)
(iii) If every price rises by Rs 5, by how much does each day's revenue increase? (1 mark)

Try it first, then show the solutionSolution

(i) $S = 403550302520203025$ ( $3 \times 3$ ), $P = 152030$ ( $3 \times 1$ ), so $S P$ is $3 \times 1$ . (1 mark)
(ii) Day 1: $600 + 600 + 600 = 1800$ . Day 2: $525 + 500 + 900 = 1925$ . Day 3: $750 + 400 + 750 = 1900$ . (2 marks)
(iii) New prices $P + 5111$ , so the increase is $5 \times$ (row sums of S): $5 \times 90 = 450$ , $5 \times 90 = 450$ , $5 \times 95 = 475$ . (1 mark)

Answer: (i) $S P$ is $3 \times 1$ . (ii) Rs 1800, Rs 1925, Rs 1900 (total Rs 5625). (iii) Rs 450, Rs 450 and Rs 475.

Where students lose marks in Multiplication of Matrices

The slips examiners see most often, and what to write instead.

Mistake: Multiplying entry by entry
Matrix product is row-by-column, not position-by-position.
Wrong working
$[1324] [5768] = [5211232]$ .
Right working
$= [5 + 14 15 + 28 6 + 16 18 + 32] = [19432250]$ .
Do this: Write each entry as a sum of products before simplifying.
Usually costs: Every mark that depends on the product.
Mistake: Expanding (A + B)² as A² + 2AB + B²
$A B$ and $B A$ are usually different, so they cannot be combined.
Wrong working
$(A + B)^{2} = A^{2} + 2 A B + B^{2}$ .
Right working
$(A + B)^{2} = A^{2} + A B + B A + B^{2}$ .
Do this: Expand brackets keeping the left-right order of every product.
Usually costs: The method marks in identity-type questions.
Mistake: Writing a number instead of a multiple of I
You cannot add a number to a matrix.
Wrong working
$A^{2} - 6 A + 5 = [7 - 12 + 5 \dots 18 - 18 + 5]$ .
Right working
$A^{2} - 6 A + 5 I$ : the 5 goes only on the diagonal, giving $[0000]$ .
Do this: Replace the constant $c$ by $c I$ before computing.
Usually costs: The final answer mark.
Mistake: Cancelling A from AB = AC
Zero products exist, so A cannot be cancelled unless it is invertible.
Wrong working
$A B = A C \Rightarrow B = C$ .
Right working
$A = [1111]$ , $B = O$ , $C = [1 - 1 - 1 1]$ : $A B = A C = O$ but $B \neq = C$ .
Do this: Cancel only after multiplying by $A^{- 1}$ , and only when $A^{- 1}$ exists.
Usually costs: A full assertion-reason or MCQ.

Exam tips

Conformability and order MCQs on matrix products appeared in the 2024 and 2025 board papers; write the orders before choosing.
In 'show that $A^{2} - k A + m I = O$ ' questions, display $A^{2}$ , $k A$ and $m I$ as separate matrices; this earns method marks even if one entry slips.
For case studies, label rows and columns of each matrix (days, items, shops) so the product's meaning is clear.
When finding unknowns, multiply from left to right; never change the order of the matrices.
Check one entry of a long product by an independent quick estimate; sign errors are the most common slip.

Practice questions on Multiplication of Matrices

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If A is a $3 \times 4$ matrix and B is a $4 \times 2$ matrix, then:
(a) AB is $3 \times 2$ and BA is not defined
(b) AB is $4 \times 4$
(c) both AB and BA are defined
(d) AB is not defined
Hint
Match the inner numbers for each order of multiplication.
Answer
(a). For AB: $(3 \times 4) (4 \times 2)$ gives $3 \times 2$ . For BA: $(4 \times 2) (3 \times 4)$ , and $2 \neq = 3$ , so BA is not defined.
Q2 · 1 mark · MCQBoard level
If $A = [0010]$ , then $A^{2}$ is:
(a) $A$
(b) $I$
(c) $O$
(d) $2 A$
Hint
Multiply row by column; every product has a zero factor.
Answer
(c) $O$ , although $A \neq = O$ .
Q3 · 2 marks · Very short answerBoard level
If $A = [21 - 1 3]$ and $B = [1 - 2 40]$ , find $A B$ and $B A$ . Is $A B = B A$ ?
Hint
Compute each entry with its full sum.
Answer
$A B = [2 + 2 1 - 6 8 + 0 4 + 0] = [4 - 5 84]$ ; $B A = [2 + 4 - 4 + 0 - 1 + 12 2 + 0] = [6 - 4 112]$ . So $A B \neq = B A$ .
Q4 · 3 marks · Short answerHarder board level
If $A = [1201]$ , find $A^{2}$ and $A^{3}$ , guess $A^{n}$ , and write $A^{4}$ .
Hint
Watch the bottom-left entry as the power grows.
Answer
$A^{2} = [1401]$ , $A^{3} = [1601]$ , so $A^{n} = [1 2 n 01]$ and $A^{4} = [1801]$ .
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): For any two square matrices A and B of the same order, $(A + B) (A - B) = A^{2} - B^{2}$ .
Reason (R): Matrix multiplication is not commutative in general.
(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.
Hint
Expand $(A + B) (A - B)$ keeping the order of each product.
Answer
(d). $(A + B) (A - B) = A^{2} - A B + B A - B^{2}$ , which equals $A^{2} - B^{2}$ only if $A B = B A$ . So A is false in general; R is true.
Q6 · 3 marks · Short answerHarder board level
If $A = [1221]$ , show that $A^{2} - 2 A - 3 I = O$ . Hence express $A^{3}$ in the form $p A + q I$ .
Hint
From the relation, $A^{2} = 2 A + 3 I$ ; multiply both sides by A.
Answer
$A^{2} = [5445]$ and $2 A + 3 I = [5445]$ , so $A^{2} - 2 A - 3 I = O$ . Then $A^{3} = 2 A^{2} + 3 A = 2 (2 A + 3 I) + 3 A = 7 A + 6 I$ . Check: $A^{3} = [13141413] = 7 A + 6 I$ .
Q7 · 4 marks · Case-basedHOTS / competency
Case study: Suppose two shops sell school kits. Shop P sells 3 notebooks, 2 pens and 1 geometry box in a kit; shop Q sells 2 notebooks, 4 pens and 2 geometry boxes. A notebook costs Rs 60, a pen Rs 40 and a geometry box Rs 100.
(i) Write the contents as a $2 \times 3$ matrix K and the prices as a column matrix C. (1 mark)
(ii) Find KC and interpret it. (2 marks)
(iii) Can CK be found? Give its order if so. (1 mark)
Hint
Kit price = sum of (number of items × price of item).
Answer
(i) $K = [322412]$ , $C = 6040100$ . (ii) $K C = [180 + 80 + 100 120 + 160 + 200] = [360480]$ : a kit costs Rs 360 at P and Rs 480 at Q. (iii) C is $3 \times 1$ and K is $2 \times 3$ ; inner numbers 1 and 2 differ, so CK is not defined.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.When is the product AB defined?Show answer
When the number of columns of A equals the number of rows of B.
2.If A is $2 \times 3$ and B is $3 \times 4$ , what is the order of AB?Show answer
$2 \times 4$ .
3.Expand $(A + B)^{2}$ for square matrices.Show answer
$A^{2} + A B + B A + B^{2}$ .
4.Can $A B = O$ with $A \neq = O$ and $B \neq = O$ ?Show answer
Yes, for example $[1111] [1 - 1 - 1 1] = O$ .
5.What is $A I$ for a square matrix A?Show answer
$A$ (and $I A = A$ too).
6.If $A^{2} = 3 A - 2 I$ , write $A^{3}$ in terms of A and I.Show answer
$A^{3} = 3 A^{2} - 2 A = 3 (3 A - 2 I) - 2 A = 7 A - 6 I$ .

Multiplication of Matrices: frequently asked questions

Why is matrix multiplication not commutative?

Because AB uses the rows of A with the columns of B, while BA uses the rows of B with the columns of A, which are different calculations. Sometimes only one product is defined, and even for square matrices of the same order the two products usually differ.

How do I check whether two matrices can be multiplied?

Write their orders side by side, for example (2 × 3)(3 × 4). If the middle numbers are equal, the product exists, and its order is given by the outer numbers, here 2 × 4.

Is A² the same as squaring each entry?

No. A² means A multiplied by A using the row-by-column rule. Squaring each entry gives a different matrix in general.

Which NCERT section covers multiplication of matrices?

Section 3.4.5 of Chapter 3, Matrices, in NCERT Mathematics Part I for Class 12, including the properties of multiplication and non-commutativity.

Are elementary row operations in the 2026-27 syllabus?

No. Elementary operations and finding an inverse by them were dropped. Inverses are found by the adjoint method (Determinants) or from a matrix equation such as A² − 5A + 7I = O.