NDA Maths · Matrices & Determinants

Matrices: Order, Algebra, Powers & Equations

A matrix is a rectangular array of numbers; you add, scalar-multiply, multiply, transpose, raise to powers, and solve matrix equations — all governed by conformability and the fact that AB ≠ BA.

All Matrices & Determinants notes NDA Maths strategy

Why this matters

Thirty-three PYQs, mostly EASY–MODERATE — the foundation the whole chapter stands on. Questions test matrix order and multiplication conformability, counting matrices, powers and matrix polynomials (A² − kA − I = O), and the algebra traps that catch students who assume matrices behave like numbers. Master the eight concepts below and you bank the easy marks and stop losing the trap ones.

Concept 1 of 8

What a matrix is — order and types

Intuition

A matrix is just a grid of numbers arranged in rows and columns. Its order is (rows × columns), read rows-first. The shape decides almost everything — which matrices can be added, which can be multiplied, and which can have a determinant or inverse.

Definition

A matrix of order $m \times n$ has $m$ rows and $n$ columns. Key types:

Row / column matrix: a single row ( $1\times n$ ) or single column ( $m\times 1$ ).
Square matrix: $m = n$ — only these have a determinant and (possibly) an inverse.
**Null (zero) matrix $O$ :** every entry 0.
Diagonal / scalar / identity: square matrices with entries only on the main diagonal (identity $I$ has 1s there).

Worked example

A matrix has 12 entries. What orders are possible for it?

The order $m \times n$ must satisfy $mn = 12$ .
Factor pairs of 12: $1\times12,\ 2\times6,\ 3\times4,\ 4\times3,\ 6\times2,\ 12\times1$ .
That is 6 possible orders.

Answer:6 possible orders (one per ordered factor pair of 12).

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Order of a matrix with 3 rows and 5 columns?
2.
How many entries in a $4 \times 3$ matrix?
3.
Which matrices can have a determinant?
4.
Possible orders for a matrix with 7 entries?

Concept 2 of 8

Equality, addition, and scalar multiplication

Intuition

Two matrices are equal only if they have the same order AND every corresponding entry matches — which turns a matrix equation into a set of ordinary equations. Addition and scalar multiplication are entry-by-entry, so they need the same order.

Definition

Matrices are equal iff same order and $a_{ij} = b_{ij}$ for all $i, j$ . Addition is defined only for the same order, done entrywise. Scalar multiplication $kA$ multiplies every entry by $k$ . Equating matrices entrywise is the standard way to solve for unknown entries.

Worked example

3A - B = \begin{pmatrix} 4 & 5 \\ 9 & 1 \end{pmatrix}

, where

A = \begin{pmatrix} x & 2 \\ 3 & 1 \end{pmatrix}

and

B = \begin{pmatrix} 2 & 1 \\ 0 & y \end{pmatrix}

, find

x

and

y

Scalar-multiply: $3A = \begin{pmatrix} 3x & 6 \\ 9 & 3 \end{pmatrix}$ .
Subtract entrywise: $3A - B = \begin{pmatrix} 3x - 2 & 5 \\ 9 & 3 - y \end{pmatrix}$ .
Equate with $\begin{pmatrix} 4 & 5 \\ 9 & 1 \end{pmatrix}$ : $3x - 2 = 4$ and $3 - y = 1$ .
Solve: $x = 2,\ y = 2$ . (The off-diagonal entries 5 and 9 already match — a built-in consistency check.)

Answer:

x = 2,\ y = 2

Practice this conceptself-check · 4 quick reps

Try it yourself

2\begin{pmatrix} a & 1 \\ 0 & b \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 2 \\ 0 & 7 \end{pmatrix}

, find

a

and

b

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Can you add a $2\times3$ and a $3\times2$ matrix?
2.
$3\begin{pmatrix}1&0\\2&1\end{pmatrix}$ — top-left entry?
3.
If $\begin{pmatrix}x\\2\end{pmatrix} = \begin{pmatrix}5\\y\end{pmatrix}$ , find x, y.
4.
Two matrices are equal when?

From the bank · past-year question

Example 2Matrices & DeterminantsMODERATE

A = \begin{pmatrix}4i-6 & 10i \\ 14i & 6+4i\end{pmatrix}

and

k = \dfrac{1}{2i}

, where

i = \sqrt{-1}

, then

kA

is equal to

[Q11 · Sep · 2017]

Concept 3 of 8

Matrix multiplication and conformability

Intuition

To multiply

A

B

, the inner dimensions must match: an

m\times n

times an

n\times p

gives an

m\times p

. Each entry of the product is a row of

A

dotted with a column of

B

. Crucially, matrix multiplication is not commutative —

AB

and

BA

can differ, or one may not even exist.

Definition

$A_{m\times n}\,B_{n\times p} = (AB)_{m\times p}$ ; the product is defined only when $A$ 's column count equals $B$ 's row count. Entry $(AB)_{ij} = \sum_k a_{ik}b_{kj}$ . In general $AB \neq BA$ . The quadratic form $[x\ y\ z]\,M\,[x\ y\ z]^T$ is $1\times1$ (a scalar).

Order of a product

A_{m\times n}\, B_{n\times p} = (AB)_{m\times p}

Worked example

A

4\times3

and

B

3\times2

, find the orders of

AB

and

BA

(or state if undefined).

$AB$ : inner dims $3 = 3$ match → order = outer dims = $4\times2$ .
$BA$ : inner dims would be 2 ( $B$ 's columns) vs 4 ( $A$ 's rows) — $2 \neq 4$ , so $BA$ is undefined.
$AB$ exists but $BA$ does not — another way $AB \neq BA$ .

Answer:

AB

4\times2

;

BA

is undefined.

Practice this conceptself-check · 4 quick reps

Try it yourself

A

2\times3

and

AB

2\times5

, what is the order of

B

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Order of $(2\times4)(4\times1)$ ?
2.
Can you compute $(3\times2)(3\times2)$ ?
3.
Is $AB = BA$ in general?
4.
Order of $[x\ y\ z]_{1\times3}\,M_{3\times3}\,[x\ y\ z]^T_{3\times1}$ ?

From the bank · past-year question

Example 3Matrices & DeterminantsEASY

A

is a matrix of order

3\times5

and

B

is a matrix of order

5\times3

, then the order of

AB

and

BA

will respectively be

[Q41 · Apr · 2020]

Drill 12 more on matrix multiplication and conformability

Concept 4 of 8

Counting matrices

Intuition

If each of the

N

entries of a matrix can independently take one of

k

values, there are

k^N

matrices. If instead you're asked how many ORDERS a matrix with

N

entries can have, count the factor pairs of

N

Definition

By entries: an $m\times n$ matrix has $mn$ entries; if each is chosen from a set of $k$ values, there are $k^{mn}$ such matrices. By order: the number of possible orders for a matrix with exactly $N$ entries equals the number of (ordered) factor pairs of $N$ , i.e. the number of divisors of $N$ .

Worked example

How many distinct

2\times2

matrices have every entry from

\{1, 2\}

A $2\times2$ matrix has 4 entries.
Each entry independently is 1 or 2 → 2 choices each.
Total $= 2^4 = 16$ .

Answer:16 matrices.

Practice this conceptself-check · 4 quick reps

Try it yourself

Using all the prime numbers less than 30 as entries (each used once), how many different ORDERS of matrix are possible?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Number of $2\times2$ matrices with entries from $\{0,1\}$ ?
2.
Number of $3\times3$ matrices with entries 0 or 1?
3.
Orders possible for a matrix with 6 entries?
4.
Number of $2\times2$ matrices with entries from $\{1,2,3,4\}$ ?

From the bank · past-year question

Example 4Matrices & DeterminantsEASY

How many distinct

2\times2

matrices exist with all four entries taken from

\{1,2\}

[Q47 · Sep · 2021]

Drill 2 more on counting matrices

Concept 5 of 8

Transpose and its rules

Intuition

The transpose

A^T

(or

A'

) flips rows into columns. The one rule that trips people up is the reversal law: the transpose of a product reverses the order —

(AB)^T = B^T A^T

Definition

$(A^T)_{ij} = a_{ji}$ . Rules: $(A^T)^T = A$ ; $(A+B)^T = A^T + B^T$ ; $(kA)^T = kA^T$ ; and ** $(AB)^T = B^T A^T$ ** (order reverses). $A^T$ has the transposed order: an $m\times n$ matrix transposes to $n\times m$ .

Transpose of a product (reversal law)

(AB)^T = B^T A^T

Worked example

Which of these are correct for matrices

A, B, C

of the same order: (1)

(A+B+C)^T = A^T+B^T+C^T

; (2)

(AB)^T = A^T B^T

(1) Transpose distributes over addition → TRUE.
(2) Transpose of a product REVERSES order: $(AB)^T = B^T A^T$ , not $A^T B^T$ → FALSE.

Answer:Only (1) is correct; (2) violates the reversal law.

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$(A^T)^T = ?$
2.
$(AB)^T = ?$
3.
Transpose order of a $2\times5$ matrix?
4.
$(A + B)^T = ?$

From the bank · past-year question

Example 5Matrices & DeterminantsMODERATE

Consider the following in respect of matrices A, B and C of same order: 1.

(A+B+C)' = A' + B' + C'

(AB)' = A'B'

(ABC)' = C'B'A'

where A' is the transpose of the matrix A. Which of the above are correct?

[Q28 · Sep · 2018]

Concept 6 of 8

Powers of a matrix

Intuition

A^n

means

A

multiplied by itself

n

times (only square matrices). The exam trick is to compute

A^2

, spot a pattern (it repeats, or equals a multiple of

A

I

), and ride that pattern instead of grinding all

n

products.

Definition

For square $A$ : $A^2 = A\cdot A$ , $A^n = A^{n-1}A$ . Look for cycles: e.g. a swap matrix $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ squares to $I$ , so even powers are $I$ . A rotation by $\theta$ to the $n$ th power is rotation by $n\theta$ .

Worked example

A = \begin{pmatrix}1&1\\0&1\end{pmatrix}

, find

A^3

$A^2 = \begin{pmatrix}1&1\\0&1\end{pmatrix}\begin{pmatrix}1&1\\0&1\end{pmatrix} = \begin{pmatrix}1&2\\0&1\end{pmatrix}$ .
$A^3 = A^2 A = \begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}1&1\\0&1\end{pmatrix} = \begin{pmatrix}1&3\\0&1\end{pmatrix}$ .
(Pattern: $A^n = \begin{pmatrix}1&n\\0&1\end{pmatrix}$ .)

Answer:

A^3 = \begin{pmatrix}1&3\\0&1\end{pmatrix}

Practice this conceptself-check · 4 quick reps

Try it yourself

A = \begin{pmatrix}-2&2\\2&-2\end{pmatrix}

, express

A^2

in terms of

A

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$I^n = ?$
2.
If $A^2 = I$ , what is $A^{10}$ ?
3.
Rotation by $\theta$ , raised to power $n$ , is rotation by?
4.
If $A^2 = A$ , what is $A^5$ ?

From the bank · past-year question

Example 6Matrices & DeterminantsEASY

A = \begin{pmatrix}0&1\\1&0\end{pmatrix}

, then the value of

A^4

[Q38 · Apr · 2017]

Drill 3 more on powers of a matrix

Concept 7 of 8

Matrix polynomials and equations

Intuition

A matrix can satisfy a polynomial equation like

A^2 - kA - I = O

. Compute

A^2

, write the equation entrywise (or use the Cayley–Hamilton shortcut for

2\times2

: every matrix satisfies

A^2 - (\text{trace})A + (\det)I = O

), and read off the unknown.

Definition

Expressions like $A^2 - 4A$ , $23A^3 - 19A^2 - 4A$ are evaluated by substituting powers of $A$ . For a $2\times2$ matrix, Cayley–Hamilton gives $A^2 - (a+d)A + (ad-bc)I = O$ where $a+d$ is the trace and $ad-bc$ the determinant — the fastest route to the constant $k$ in $A^2 - kA + cI = O$ problems.

Cayley–Hamilton (2×2)

A^2 - (a+d)\,A + (ad - bc)\,I = O

Worked example

A = \begin{pmatrix}3&1\\2&4\end{pmatrix}

and

A^2 - kA + 10I_2 = O

, find

k

Trace $= 3 + 4 = 7$ ; determinant $= 3\cdot4 - 1\cdot2 = 10$ .
Cayley–Hamilton: $A^2 - (\text{trace})A + (\det)I = O$ , i.e. $A^2 - 7A + 10I = O$ .
Compare with $A^2 - kA + 10I = O$ : $k = 7$ .

Answer:

k = 7

(the trace).

Practice this conceptself-check · 4 quick reps

Try it yourself

A = \begin{pmatrix}1&2&2\\2&1&2\\2&2&1\end{pmatrix}

, find

A^2 - 4A

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Cayley–Hamilton constant in $A^2 - kA + \dots$ for 2×2 — k is the?
2.
If $A = I$ , then $A^2 - 4A$ equals?
3.
If $A^2 = 5I$ , what is $A^4$ ?
4.
Trace of $\begin{pmatrix}2&1\\3&5\end{pmatrix}$ ?

From the bank · past-year question

Example 7Matrices & DeterminantsMODERATE

A = \begin{pmatrix}1 & 2 \\ 2 & 3\end{pmatrix}

and

A^2 - kA - I_2 = O

, then what is k?

[Q26 · Apr · 2018]

Drill 6 more on matrix polynomials and equations

Concept 8 of 8

Matrix algebra — where numbers' rules break

Intuition

Matrices look like numbers but don't always obey number-algebra. Three classics:

AB = O

does NOT force

A

B

to be zero;

(A+B)(A-B) \neq A^2 - B^2

unless

A, B

commute; and

(A+B)^2 \neq A^2 + 2AB + B^2

for the same reason.

Definition

Because $AB \neq BA$ in general:

$(A+B)(A-B) = A^2 - AB + BA - B^2$ , which equals $A^2 - B^2$ only if $AB = BA$ .
$(A+B)^2 = A^2 + AB + BA + B^2$ .
$AB = O$ is possible with both $A \neq O$ and $B \neq O$ (zero divisors exist).

Transpose/adjoint/inverse of a product all reverse order.

Worked example

For matrices

A, B

of the same order, is

(A+B)(A-B) = A^2 - B^2

always true?

Expand: $(A+B)(A-B) = A^2 - AB + BA - B^2$ .
This equals $A^2 - B^2$ only if $-AB + BA = O$ , i.e. $AB = BA$ .
Since matrices generally don't commute, the identity fails in general.

Answer:No — only when

A

and

B

commute (

AB = BA

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Does $AB = O$ imply $A = O$ or $B = O$ ?
2.
$(A+B)(A-B) = A^2 - B^2$ requires?
3.
$(A+B)^2$ expands to?
4.
Is $AB = BA$ generally true?

From the bank · past-year question

Example 8Matrices & DeterminantsMODERATE

Consider the following in respect of matrices A and B of same order: 1.

A^2-B^2=(A+B)(A-B)

(A-I)(I+A)=O \Leftrightarrow A^2=I

where I is the identity matrix and O is the null matrix. Which of the above is/are correct?

[Q35 · Sep · 2018]

Don't import $a^2 - b^2 = (a+b)(a-b)$ into matrices

Every number identity that secretly uses commutativity can break for matrices.

(A+B)(A-B)

(A+B)^2

(AB)^2 = A^2B^2

all FAIL unless

AB = BA

. When an option assumes one of these, it's almost always the trap answer.

Drill 3 more on matrix algebra — where numbers' rules break

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Matrix multiplication and conformability
Order of a product
$A_{m\times n}\, B_{n\times p} = (AB)_{m\times p}$
Transpose and its rules
Transpose of a product (reversal law)
$(AB)^T = B^T A^T$
Matrix polynomials and equations
Cayley–Hamilton (2×2)
$A^2 - (a+d)\,A + (ad - bc)\,I = O$

Watch out for (1)

Don't import $a^2 - b^2 = (a+b)(a-b)$ into matrices
→ Matrix algebra — where numbers' rules break

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Matrices & DeterminantsMODERATE

Let

X

be a matrix of order

3 \times 3

Y

be a matrix of order

2 \times 3

and

Z

be a matrix of order

3 \times 2

. Which of the following statements are correct? A.

(ZY)X

is defined and is a square matrix of order 3. B.

Y(XZ)

is defined and is a square matrix of order 2. C.

X(YZ)

is not defined. Select the answer using the code given below.

[Q1 · Sep · 2024]

Example 2Matrices & DeterminantsMODERATE

How many matrices of different orders are possible with elements comprising all prime numbers less than 30?

[Q12 · Apr · 2021]

Example 3Matrices & DeterminantsEASY

A = \begin{pmatrix}-2 & 2 \\ 2 & -2\end{pmatrix}

, then which one of the following is correct?

[Q98 · Sep · 2017]

Example 4Matrices & DeterminantsHARD

Let

A=\begin{pmatrix}0&2\\-2&0\end{pmatrix}

and

(mI+nA)^{2}=A

where

m

n

are positive real numbers and

I

is the identity matrix. What is

(m+n)

equal to?

[Q30 · Sep · 2021]

Example 5Matrices & DeterminantsMODERATE

Consider the following statements in respect of square matrices

A

and

B

of same order: 1. If

AB

is a null matrix, then at least one of

A

and

B

is a null matrix. 2. If

AB

is an identity matrix, then

BA=AB

. Which of the above statements is/are correct?

[Q8 · Apr · 2022]

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