MathsISC Class 12 · Matrices

Matrix Multiplication and Powers of a Matrix: Rules, Traps and Examples

Updated 15 October 2026 · Fact-checked

The product AB exists only when A has as many columns as B has rows, and then AB has the rows of A and the columns of B. Each entry is a row of A times a column of B. AB and BA usually differ, and AB can be zero with neither matrix zero.

Check that the inner orders match; the outer orders give the size of AB, and each entry is one row times one column.

Illustration drawn for QuestionBuddy.

Understand Matrix Multiplication and Powers

When is AB defined? If $A$ is $m \times n$ and $B$ is $n \times p$ , the "inner" orders match ( $n = n$ ) and $A B$ exists with order $m \times p$ , the "outer" orders. If the inner orders differ, the product is not defined. So $A B$ may exist while $B A$ does not: a $2 \times 3$ matrix times a $3 \times 1$ matrix is fine, but the reverse is not.

How to multiply. The $(i, j)$ entry of $A B$ is row $i$ of $A$ multiplied term by term with column $j$ of $B$ , and added: $(A B)_{ij} = a_{i 1} b_{1 j} + a_{i 2} b_{2 j} + \dots + a_{in} b_{nj}$ . Work one entry at a time and write the products before adding; most errors here are arithmetic slips, as the ISC 2025 examiners noted.

Rules that hold. Multiplication is associative, $(A B) C = A (B C)$ , and distributive, $A (B + C) = A B + A C$ and $(A + B) C = A C + B C$ . The identity matrix behaves like 1: $A I = I A = A$ . Scalars move freely: $(k A) B = k (A B) = A (k B)$ .

Rules that fail. Three habits from ordinary algebra break down, and the syllabus asks for examples of each (with square matrices of order up to 3):

Non-commutativity: in general $A B \neq = B A$ , even when both exist.
Zero products: $A B = O$ is possible with $A \neq = O$ and $B \neq = O$ .
No cancellation: $A B = A C$ does not imply $B = C$ unless $A$ is invertible.

Because $A B \neq = B A$ , the familiar expansions change: $(A + B)^{2} = A^{2} + A B + B A + B^{2}$ , which equals $A^{2} + 2 A B + B^{2}$ only if $A B = B A$ . Likewise $(A + B) (A - B) = A^{2} - A B + B A - B^{2}$ .

Powers. For a square matrix, $A^{2} = AA$ , $A^{3} = A^{2} A$ , and so on. Powers of the same matrix always commute: $A^{m} A^{n} = A^{m + n} = A^{n} A^{m}$ . If $A B = B A$ , then $A^{n} B = B A^{n}$ for every positive integer $n$ , a fact tested in ISC 2024. Some special behaviours are worth recognising on sight:

Nilpotent: $A^{k} = O$ for some $k$ . Any $2 \times 2$ matrix of the form $[00 a 0]$ has $A^{2} = O$ , so every higher power is $O$ .
Idempotent: $A^{2} = A$ , so $A^{n} = A$ .
Involutory: $A^{2} = I$ , so even powers are $I$ and odd powers are $A$ .
Patterns: $[10 a 1]^{n} = [10 na 1]$ .

Matrix polynomials. Questions such as "show that $A^{2} - 6 A + 5 I = O$ " are evaluated term by term: compute $A^{2}$ , scale $A$ , add the multiple of $I$ . For a $2 \times 2$ matrix the coefficients are always the trace and the determinant: $A^{2} - (a + d) A + (a d - b c) I = O$ , a useful check on your arithmetic. Such an equation also gives the inverse: multiplying $A^{2} - 6 A + 5 I = O$ by $A^{- 1}$ gives $A - 6 I + 5 A^{- 1} = O$ .

Matrices for data. A table of quantities times a column of prices gives a column of totals. In ISC 2025, candidates had to write the quantities of three kinds of bag used by three shopkeepers as a $3 \times 3$ matrix and multiply by the price column; the examiners noted that some candidates did not see it as a matrix product at all.

Matrix Multiplication and Powers: formulas to remember

Compatibility: $A_{m \times n} B_{n \times p} = (A B)_{m \times p}$; Inner orders must match; outer orders give the product.
Entry of a 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: $(A B) C = A (B C), A (B + C) = A B + A C$; Brackets can move; factors cannot swap.
Non-commutative square: $(A + B)^{2} = A^{2} + A B + B A + B^{2}$; Equals A² + 2AB + B² only when AB = BA.
Commuting powers: $A B = B A \Rightarrow A^{n} B = B A^{n}$; Proved by moving B past one A at a time.
Upper unitriangular power: $[10 a 1]^{n} = [10 na 1]$; Check n = 2 to see the pattern.
2 × 2 matrix equation: $A^{2} - (a + d) A + (a d - b c) I = O$; For A = [a b; c d]. A quick check on matrix-polynomial questions.

How to solve Matrix Multiplication and Powers questions

Use this routine for products, powers and matrix-polynomial questions.

1Write the orders under each matrix and check that the inner orders match. State the order of the product.
2Compute the product entry by entry: for each row of the first matrix, multiply by each column of the second. Write each sum before simplifying.
3For $A^{2}$ or $A^{3}$ , multiply step by step ( $A^{3} = A^{2} \cdot A$ ) and look for a pattern (zero rows, identity, a multiple of $A$ ).
4For a matrix polynomial such as $A^{2} - 5 A + 7 I$ , compute each term as a separate matrix, then combine entry by entry.
5For "find $k$ " questions ( $A^{2} = k A + c I$ ), compare one off-diagonal entry to find $k$ , then confirm with a diagonal entry.
6For data questions, put the items being counted along the rows of one matrix and the matching prices or weights in a column, so that the inner orders match.
7Check one entry of your answer by a second method (for example, the trace-determinant equation for $2 \times 2$ matrices).

Matrix Multiplication and Powers 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.

Matrix product for totals

Suppose two school canteens place a daily order with a caterer. Canteen 1 orders 40 samosas, 30 sandwiches and 50 juices; canteen 2 orders 60 samosas, 20 sandwiches and 40 juices. A samosa costs Rs 12, a sandwich Rs 25 and a juice Rs 15.

Write the orders as a matrix $Q$ and the prices as a column $P$ , find each canteen's bill using $QP$ , and explain why $P Q$ is not defined.

Try it first, then show the solutionSolution

$Q = [406030205040]$ is $2 \times 3$ ; $P = 122515$ is $3 \times 1$ . Inner orders match, so $QP$ is $2 \times 1$ .
Canteen 1: $40 (12) + 30 (25) + 50 (15) = 480 + 750 + 750 = 1980$ .
Canteen 2: $60 (12) + 20 (25) + 40 (15) = 720 + 500 + 600 = 1820$ .
$P Q$ would be $(3 \times 1) (2 \times 3)$ : the inner orders 1 and 2 differ, so it is not defined.

Answer: $QP = [19801820]$ : Rs 1980 and Rs 1820. $P Q$ is not defined.

Memory hook: Quantities times prices: rows of items meet a column of prices, and the bill drops out.

Non-commutativity of matrix multiplication

Imagine a photo-editing app that moves points of a picture using matrices. Rotating by $9 0^{\circ}$ anticlockwise is $R = [01 - 1 0]$ and flipping in the x-axis is $F = [10 0 - 1]$ .

Where does the point $(2, 1)$ go if the app rotates then flips? If it flips then rotates? Find $F R$ and $R F$ .

Try it first, then show the solutionSolution

Rotate then flip means $F (R p)$ . $R p = [- 1 2]$ , then $F (R p) = [- 1 - 2]$ .
Flip then rotate means $R (F p)$ . $F p = [2 - 1]$ , then $R (F p) = [12]$ .
$F R = [0 - 1 - 1 0]$ and $R F = [0110]$ , which are different.

Answer: Rotate then flip sends $(2, 1)$ to $(- 1, - 2)$ ; flip then rotate sends it to $(1, 2)$ . $F R \neq = R F$ .

Memory hook: Rotate-then-flip is not flip-then-rotate: the order of matrices matters.

Tricks to remember Matrix Multiplication and Powers

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

Inner match, outer order
Write the orders side by side: (m × n)(n × p). Inner numbers must match; outer numbers give the answer.
Use it when: Every "is AB defined / what is its order" question.
Row runs, column falls
Run your left finger along a row of A while your right finger falls down a column of B.
Use it when: Avoiding mixed-up entries when multiplying.
Three broken rules
Not commutative, zero products possible, no cancellation: "NZN".
Use it when: Assertion-reason items on properties of multiplication.
Zero above, zero after
A matrix with zeros on and below the diagonal becomes O after enough multiplications (order 2: square; order 3: cube).
Use it when: Spotting nilpotent matrices such as the ISC 2025 MCQ on A¹⁶.

Matrix Multiplication and Powers: solved examples

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

Example 1: Orders of products · 1 mark · MCQBoard level

If $A$ is a $2 \times 3$ matrix and $B$ is a $3 \times 2$ matrix, the orders of $A B$ and $B A$ are:
(a) $2 \times 2$ and $3 \times 3$
(b) $3 \times 3$ and $2 \times 2$
(c) both $2 \times 2$
(d) $A B$ is not defined

Try it first, then show the solutionSolution

$(2 \times 3) (3 \times 2)$ gives $2 \times 2$ ; $(3 \times 2) (2 \times 3)$ gives $3 \times 3$ .

Answer: (a) $2 \times 2$ and $3 \times 3$ .

Example 2: A matrix equation and the inverse · 3 marks · Short answerBoard level

For $A = [2134]$ , show that $A^{2} - 6 A + 5 I = O$ and hence find $A^{- 1}$ .

Try it first, then show the solutionSolution

$A^{2} = [2134] [2134] = [4 + 3 2 + 4 6 + 12 3 + 16] = [761819]$ . (1 mark)
$A^{2} - 6 A + 5 I = [7 - 12 + 5 6 - 6 18 - 18 19 - 24 + 5] = [0000]$ . (1 mark)
Multiply by $A^{- 1}$ (it exists, since $∣ A ∣ = 5 \neq = 0$ ): $A - 6 I + 5 A^{- 1} = O$ , so $A^{- 1} = \frac{1}{5} (6 I - A) = \frac{1}{5} [4 - 1 - 3 2]$ . (1 mark)

Answer: $A^{2} - 6 A + 5 I = O$ and $A^{- 1} = \frac{1}{5} [4 - 1 - 3 2]$ .

Example 3: A zero product and non-commutativity · 2 marks · Short answerHarder board level

Let $A = [0012]$ and $B = [1000]$ . Find $A B$ and $B A$ . What do the results show?

Try it first, then show the solutionSolution

$A B = [0 + 0 0 + 0 0 + 0 0 + 0] = [0000]$ . (½ mark)
$B A = [0010]$ . (½ mark)
Neither $A$ nor $B$ is the zero matrix, yet $A B = O$ : zero products are possible. And $A B \neq = B A$ : multiplication is not commutative. (1 mark)

Answer: $A B = O$ but $B A = [0010]$ .

Example 4: Spotting a pattern in powers · 3 marks · Short answerHarder board level

If $A = [1021]$ , find $A^{2}$ and $A^{3}$ , write $A^{n}$ , and hence find $A^{10}$ .

Try it first, then show the solutionSolution

$A^{2} = [10 2 + 2 1] = [1041]$ . (1 mark)
$A^{3} = A^{2} A = [10 2 + 4 1] = [1061]$ . (1 mark)
Each multiplication by $A$ adds 2 to the top-right entry, so $A^{n} = [10 2 n 1]$ and $A^{10} = [10201]$ . (1 mark)

Answer: $A^{n} = [10 2 n 1]$ ; $A^{10} = [10201]$ .

Example 5: A nilpotent 3 × 3 matrix · 1 mark · MCQHarder board level

If $N = 000200530$ , then $N^{3}$ is:
(a) $I$
(b) $N$
(c) $O$
(d) $6 N$

Try it first, then show the solutionSolution

$N^{2} = 000000600$ ; multiplying once more by $N$ gives the zero matrix, because the only non-zero entry of $N^{2}$ meets the zero bottom row of $N$ .

Answer: (c) $O$ .

Example 6: Revenue and profit of three stalls · 5 marks · Case-basedHOTS / competency

Suppose three food stalls at a fair sell tea, coffee and juice. The cups sold are given by $Q = 302540203010101520$ (rows: stalls 1, 2, 3; columns: tea, coffee, juice). Selling prices are Rs 10, 15, 20 a cup and costs are Rs 6, 9, 12 a cup.
(i) Find each stall's revenue. (2 marks)
(ii) Find each stall's profit and the total profit. (2 marks)
(iii) If every selling price rises by 10%, write the new revenue column without multiplying again. (1 mark)

Try it first, then show the solutionSolution

(i) $Q 101520 = 300 + 300 + 200 250 + 450 + 300 400 + 150 + 400 = 8001000950$ .
(ii) Profit per cup $= 468$ . $Q 468 = 120 + 120 + 80 100 + 180 + 120 160 + 60 + 160 = 320400380$ . Total $= 1100$ .
(iii) New prices are $1.1$ times the old, so revenue $= Q (1.1 P) = 1.1 (QP) = 88011001045$ .

Answer: Revenues Rs 800, 1000, 950; profits Rs 320, 400, 380 (total Rs 1100); new revenues Rs 880, 1100, 1045.

Where students lose marks in Matrix Multiplication and Powers

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

Mistake: Using (A + B)² = A² + 2AB + B²
This assumes AB = BA, which is not true in general.
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 as products, keeping the order of every factor.
Usually costs: The whole part in a proof or simplification.
Mistake: Arithmetic slips in multiplication
The ISC 2025 examiners reported basic multiplication errors on a simple power question.
Do this: Write each entry as a sum of products (for example 2·3 + 3·4) before simplifying, and check one entry by another route.
Usually costs: 1 mark in Section A; more in longer questions.
Mistake: Cancelling a matrix
AB = AC does not give B = C unless A is invertible.
Wrong working
$A B = A C \Rightarrow B = C$ .
Right working
$A B = A C$ and $∣ A ∣ \neq = 0 \Rightarrow A^{- 1} A B = A^{- 1} A C \Rightarrow B = C$ .
Do this: Only "cancel" by multiplying both sides by an inverse, and say that it exists.
Usually costs: 1 to 2 marks.
Mistake: Not recognising data as a matrix product
In ISC 2025 some candidates could not write the shopkeepers' bag data as a matrix multiplication.
Do this: Put the counts in a matrix (one row per person) and the prices in a column; check the inner orders match.
Usually costs: Both marks of a 2-mark data question.

Exam tips

Write the orders under the matrices before multiplying; it prevents undefined products and tells you the shape of the answer.
For powers, compute $A^{2}$ carefully, then look for a pattern before multiplying further. ISC MCQs often hide a nilpotent or involutory matrix.
In matrix-polynomial questions, show $A^{2}$ , the multiple of $A$ and the multiple of $I$ separately, then combine.
When an assertion-reason item involves $A^{n} B = B A^{n}$ , first check whether $A B = B A$ by direct multiplication.
In a data question, label the rows and columns of each matrix in words; it shows the examiner your matrix is set up correctly.

Practice questions on Matrix Multiplication and Powers

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 $A = [10 0 - 1]$ , then $A^{50}$ equals:
(a) $A$
(b) $I$
(c) $- I$
(d) $O$
Hint
Find $A^{2}$ first.
Answer
(b) $I$ , since $A^{2} = I$ and 50 is even.
Q2 · 2 marks · Short answerBoard level
Find $x$ and $y$ if $[x 2 3 y] [12] = [810]$ .
Hint
Multiply out and compare entries.
Answer
$x + 6 = 8 \Rightarrow x = 2$ ; $2 + 2 y = 10 \Rightarrow y = 4$ .
Q3 · 2 marks · Short answerHarder board level
Show that $A B \neq = B A$ for $A = [25 - 1 3]$ and $B = [0110]$ , and describe what multiplying by $B$ does on each side.
Hint
Look at what happens to rows and columns.
Answer
$A B = [- 1 3 25]$ (columns of $A$ swapped); $B A = [52 3 - 1]$ (rows of $A$ swapped). They differ.
Q4 · 2 marks · Short answerHarder board level
If $A = [1221]$ and $A^{2} = k A + 3 I$ , find $k$ .
Hint
Compare an off-diagonal entry.
Answer
$A^{2} = [5445]$ . Off-diagonal: $4 = 2 k \Rightarrow k = 2$ . Check diagonal: $5 = 2 (1) + 3$ .
Q5 · 1 mark · MCQHarder board level
If $A$ is a $3 \times 4$ matrix and $B$ is a matrix such that both $A^{'} B$ and $B A^{'}$ are defined, then the order of $B$ is:
(a) $3 \times 4$
(b) $4 \times 3$
(c) $3 \times 3$
(d) $4 \times 4$
Hint
$A^{'}$ is $4 \times 3$ .
Answer
(a) $3 \times 4$ : $A^{'} B$ needs 3 rows in $B$ , and $B A^{'}$ needs 4 columns in $B$ .
Q6 · 3 marks · Short answerHOTS / competency
If $A = [cos θ - sin θ sin θ cos θ]$ , show that $A^{2} = [cos 2 θ - sin 2 θ sin 2 θ cos 2 θ]$ , and find the value of $θ$ in $(0, 2 π)$ for which $A^{2} = I$ .
Hint
Use double-angle formulas; then you need $cos 2 θ = 1$ and $sin 2 θ = 0$ .
Answer
Top-left: $cos^{2} θ - sin^{2} θ = cos 2 θ$ ; top-right: $2 sin θ cos θ = sin 2 θ$ ; similarly for the bottom row. $A^{2} = I$ needs $2 θ$ to be a multiple of $2 π$ , so $θ = π$ in $(0, 2 π)$ .
Q7 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a district has a town and its surrounding villages. Each year 10% of town residents move to the villages and 20% of village residents move to the town; nobody else moves. With $M = [0.9 0.1 0.2 0.8]$ , the populations after one year are $M X$ , where $X = [50003000]$ (town, villages) now.
(i) Find the populations after one year. (1 mark)
(ii) Find $M^{2}$ and use it to find the populations after two years. (2 marks)
(iii) Explain why the total population stays 8000. (1 mark)
Hint
Look at the column sums of $M$ .
Answer
(i) $M X = [51002900]$ . (ii) $M^{2} = [0.83 0.17 0.34 0.66]$ , $M^{2} X = [51702830]$ . (iii) Each column of $M$ adds to 1, so everyone is counted exactly once in the new totals.

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 $A B$ defined?Show answer
When the number of columns of $A$ equals the number of rows of $B$ .
2.Give the order of $A B$ if $A$ is $3 \times 2$ and $B$ is $2 \times 4$ .Show answer
$3 \times 4$ .
3.Find $A^{2}$ for $A = [0050]$ .Show answer
$O$ , the zero matrix.
4.Expand $(A + B)^{2}$ for square matrices.Show answer
$A^{2} + A B + B A + B^{2}$ .
5.If $A B = B A$ , what can you say about $A^{3} B$ ?Show answer
$A^{3} B = B A^{3}$ .
6.What is $[1031]^{n}$ ?Show answer
$[10 3 n 1]$ .

Matrix Multiplication and Powers: frequently asked questions

Why is matrix multiplication not commutative?

Because AB pairs the rows of A with the columns of B, while BA pairs the rows of B with the columns of A. These are different calculations, and they may not even give matrices of the same order. They agree only for special pairs, such as a matrix and its own powers or the identity.

Can the product of two non-zero matrices be zero?

Yes. For example, with A = [0 1; 0 2] and B = [1 0; 0 0], AB is the zero matrix. This is why you cannot conclude A = O or B = O from AB = O, unlike with numbers.

How do I find a high power of a matrix such as A¹⁶?

Compute A² and A³ and look for a pattern: a zero matrix (nilpotent), the identity (involutory), A itself (idempotent), or an entry that grows steadily. Then use the pattern instead of multiplying sixteen times.

Is the Cayley-Hamilton theorem in the ISC syllabus?

It is not named in the CISCE 2027 syllabus, but questions of the type "show that A² − 5A + 7I = O" are standard. For a 2 × 2 matrix the coefficients are the trace and determinant, which is a handy check rather than a method you need to quote.

What order of matrices can ISC questions use?

The 2027 syllabus restricts matrices to order m × n with m, n at most 3, and the examples of non-commutativity and zero products to square matrices of order up to 3.