NDA Maths · Matrices & Determinants

Special Matrices and Their Tell-Tale Properties

Named matrix types — symmetric, skew-symmetric, diagonal, orthogonal, rotation, idempotent, involutory — each carry a defining equation that instantly fixes their determinant and inverse.

All Matrices & Determinants notes NDA Maths strategy

Why this matters

Twenty-two PYQs, mostly EASY–MODERATE — high-yield because each type is recognised by one equation and then answers itself (a skew-symmetric matrix of odd order has determinant 0; an orthogonal matrix has inverse equal to its transpose). Learn the catalog and the recognition questions become instant.

Concept 1 of 6

The special-matrix catalog

Intuition

Each special matrix is defined by one equation relating

A

A^T

A^{-1}

, or a power of itself. Recognising that equation is the whole game — it pins down the determinant, the inverse, and the diagonal at a glance.

Definition

Memorise the defining property of each type; the exam tests recognition more than computation.

Type	Defining property	Key consequence
Symmetric	$A^T = A$	$a_{ij} = a_{ji}$ ; inverse (if any) is symmetric
Skew-symmetric	$A^T = -A$	diagonal all 0; odd order $\Rightarrow \det = 0$ Odd-order skew-symmetric is ALWAYS singular (det 0). Even-order need not be.
Diagonal	off-diagonal all 0	$\det =$ product of diagonal entries
Orthogonal	$AA^T = I$	$A^{-1} = A^T$ ; $\det = \pm 1$
Idempotent	$A^2 = A$	$\det \in \{0, 1\}$
Involutory	$A^2 = I$	$A^{-1} = A$ ; $\det = \pm 1$

Recognise the defining equation first; the determinant and inverse follow immediately.

Practice this conceptself-check · 4 quick reps

Try it yourself

A matrix satisfies

A^T = -A

and is of order 3. What is

\det A

, and what are its diagonal entries?

Practice — Level 1 (4 reps)

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

1.
Defining property of an orthogonal matrix?
2.
$A^2 = I$ defines which type?
3.
Diagonal entries of a skew-symmetric matrix?
4.
$\det$ of an orthogonal matrix?

From the bank · past-year question

Example 1Matrices & DeterminantsEASY

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

, then the matrix A is a/an

[Q9 · Apr · 2019]

Drill 4 more on the special-matrix catalog

Concept 2 of 6

Symmetric and skew-symmetric matrices

Intuition

Symmetric mirrors across the diagonal (

A^T = A

); skew-symmetric anti-mirrors (

A^T = -A

, forcing zero diagonal). Any square matrix splits uniquely into a symmetric part plus a skew-symmetric part — a decomposition the exam loves.

Definition

$A$ is symmetric if $A^T = A$ , skew-symmetric if $A^T = -A$ (so $a_{ii}=0$ ). Every square matrix decomposes as $A = \tfrac12(A + A^T) + \tfrac12(A - A^T)$ — symmetric part $P$ plus skew part $Q$ . Useful facts: $A + A^T$ is symmetric, $A - A^T$ is skew, $AA^T$ is symmetric; an odd-order skew-symmetric matrix has $\det = 0$ .

Symmetric + skew decomposition

A = \underbrace{\tfrac12(A + A^T)}_{\text{symmetric}} + \underbrace{\tfrac12(A - A^T)}_{\text{skew-symmetric}}

Worked example

Let

A

be a skew-symmetric matrix of order 3. Find

\det A

Skew-symmetric: $A^T = -A$ . Take determinants: $\det(A^T) = \det(-A)$ .
$\det(A^T) = \det A$ ; $\det(-A) = (-1)^3 \det A = -\det A$ .
So $\det A = -\det A \Rightarrow 2\det A = 0 \Rightarrow \det A = 0$ .

Answer:

\det A = 0

(odd-order skew-symmetric is always singular).

Practice this conceptself-check · 4 quick reps

Try it yourself

A

and

B

are symmetric matrices of the same order, is

AB

symmetric?

Practice — Level 1 (4 reps)

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

1.
$A^T = -A$ — the diagonal entries are?
2.
Det of a $3\times3$ skew-symmetric matrix?
3.
$A + A^T$ is always?
4.
$A - A^T$ is always?

From the bank · past-year question

Example 2Matrices & DeterminantsEASY

P

is a skew-symmetric matrix of order 3, then what is

\det(P)

equal to?

[Q20 · Sep · 2024]

Drill 7 more on symmetric and skew-symmetric matrices

Concept 3 of 6

Diagonal, scalar, and identity matrices

Intuition

Diagonal matrices are the friendliest: their determinant is just the product of the diagonal, their powers raise each diagonal entry to that power, and their inverse reciprocates each diagonal entry. A scalar matrix is

kI

; the identity is

1\cdot I

Definition

For a diagonal matrix $D = \text{diag}(d_1,\dots,d_n)$ : $\det D = \prod d_i$ , $D^k = \text{diag}(d_1^k,\dots,d_n^k)$ , and $D^{-1} = \text{diag}(1/d_1,\dots,1/d_n)$ (when all $d_i \neq 0$ ). A scalar matrix is $kI$ ; $\det(kI_n) = k^n$ .

Diagonal determinant and inverse

\det D = \prod_i d_i, \qquad D^{-1} = \operatorname{diag}\!\left(\tfrac{1}{d_1}, \dots, \tfrac{1}{d_n}\right)

Worked example

For

A = \begin{pmatrix}3&0&0\\0&2&0\\0&0&5\end{pmatrix}

, find

\det A

and

\det(A^2)

Diagonal → $\det A = 3\cdot2\cdot5 = 30$ .
$A^2 = \text{diag}(9, 4, 25)$ , so $\det(A^2) = 9\cdot4\cdot25 = 900 = 30^2$ .

Answer:

\det A = 30

\det(A^2) = 900

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

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

1.
$\det(\text{diag}(1,5,2))$ ?
2.
$\det(kI_3)$ ?
3.
Inverse of $\text{diag}(2,4)$ ?
4.
$(\text{diag}(2,3))^3$ ?

From the bank · past-year question

Example 3Matrices & DeterminantsEASY

A = \begin{pmatrix}2&0&0\\0&3&0\\0&0&4\end{pmatrix}

, then which of the following statements are correct? (1)

A^n

will always be singular for any positive integer

n

. (2)

A^n

will always be a diagonal matrix for any positive integer

n

. (3)

A^n

will always be a symmetric matrix for any positive integer

n

[Q9 · Sep · 2023]

Concept 4 of 6

Orthogonal matrices

Intuition

An orthogonal matrix has perpendicular unit columns, captured by

AA^T = I

. The huge payoff: its inverse is simply its transpose (no adjoint needed), and its determinant is

\pm 1

Definition

$A$ is orthogonal if $AA^T = A^T A = I$ . Then $A^{-1} = A^T$ and $\det A = \pm 1$ (since $\det(AA^T) = (\det A)^2 = 1$ ). Rotation matrices are the standard example.

Orthogonality

AA^T = I \;\Longrightarrow\; A^{-1} = A^T,\quad \det A = \pm 1

Worked example

A square matrix

A

is called orthogonal under which condition?

By definition, orthogonal means $A^T A = I$ (equivalently $AA^T = I$ ).
This is what makes $A^{-1} = A^T$ .

Answer:

A A^T = I

(equivalently

A^T = A^{-1}

Practice this conceptself-check · 4 quick reps

Try it yourself

A

is orthogonal of order 4, what is the value of

\det(A^4)

Practice — Level 1 (4 reps)

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

1.
For orthogonal $A$ , $A^{-1} = ?$
2.
$\det$ of an orthogonal matrix?
3.
Is a rotation matrix orthogonal?
4.
If $AA^T = I$ , what is $(\det A)^2$ ?

From the bank · past-year question

Example 4Matrices & DeterminantsEASY

A square matrix A is called orthogonal if (where A' is the transpose of A)

[Q4 · Sep · 2018]

Drill 4 more on orthogonal matrices

Concept 5 of 6

Rotation matrices

Intuition

The rotation matrix

R(\theta) = \begin{pmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{pmatrix}

rotates the plane. Composing rotations adds angles, so

R(\theta)R(\phi) = R(\theta+\phi)

and

R(\theta)^n = R(n\theta)

— no matrix grinding needed.

Definition

$R(\theta) = \begin{pmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{pmatrix}$ is orthogonal with $\det = 1$ . Composition law: $R(\theta)R(\phi) = R(\theta+\phi)$ ; hence $R(\theta)^n = R(n\theta)$ and $R(\theta)^{-1} = R(-\theta) = R(\theta)^T$ .

Rotation composition

R(\theta)\,R(\phi) = R(\theta + \phi), \qquad R(\theta)^n = R(n\theta)

Worked example

A = \begin{pmatrix}\cos\alpha & \sin\alpha\\-\sin\alpha & \cos\alpha\end{pmatrix}

, what is

A^2

$A$ is a rotation by $\alpha$ .
$A^2 = R(\alpha)R(\alpha) = R(2\alpha)$ .
So $A^2 = \begin{pmatrix}\cos2\alpha & \sin2\alpha\\-\sin2\alpha & \cos2\alpha\end{pmatrix}$ .

Answer:

A^2 = \begin{pmatrix}\cos2\alpha & \sin2\alpha\\-\sin2\alpha & \cos2\alpha\end{pmatrix}

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

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

1.
$R(\theta)R(\phi) = ?$
2.
$\det R(\theta)$ ?
3.
$R(\theta)^{-1} = ?$
4.
$R(30°)^3 = ?$

From the bank · past-year question

Example 5Matrices & DeterminantsEASY

A = \begin{pmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{pmatrix}

, then what is

AA^T

equal to (where

A^T

is the transpose of

A

[Q30 · Apr · 2017]

Drill 2 more on rotation matrices

Concept 6 of 6

Idempotent and involutory matrices

Intuition

An idempotent matrix is unchanged when squared (

A^2 = A

— like a projection); an involutory matrix is its own inverse (

A^2 = I

). Spot the defining square and the powers collapse instantly.

Definition

Idempotent: $A^2 = A$ (so $A^n = A$ for all $n \ge 1$ ; $\det A \in \{0,1\}$ ). Involutory: $A^2 = I$ (so $A^{-1} = A$ ; even powers are $I$ , odd powers are $A$ ). A common test matrix is the all-ones matrix $J$ , where $J^2 = nJ$ for order $n$ .

Worked example

For

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

, express

A^2

in terms of

A

, and state whether

A

is invertible.

Each entry of $A^2$ is a row of 1s dotted with a column of 1s $= 3$ . So $A^2 = 3A$ .
All rows are identical → $\det A = 0$ .
Determinant 0 → $A$ is not invertible.

Answer:

A^2 = 3A

;

A

is singular (no inverse).

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

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

1.
Idempotent means?
2.
Involutory means?
3.
If $A^2 = A$ , then $A^7 = ?$
4.
If $A^2 = I$ , then $A^{-1} = ?$

From the bank · past-year question

Example 6Matrices & DeterminantsMODERATE

Consider the following in respect of the matrix

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

: 1. Inverse of

A

does not exist 2.

A^{3}=A

3A=A^{2}

Which of the above are correct?

[Q50 · Sep · 2021]

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 (4)

Symmetric and skew-symmetric matrices
Symmetric + skew decomposition
$A = \underbrace{\tfrac12(A + A^T)}_{\text{symmetric}} + \underbrace{\tfrac12(A - A^T)}_{\text{skew-symmetric}}$
Diagonal, scalar, and identity matrices
Diagonal determinant and inverse
$\det D = \prod_i d_i, \qquad D^{-1} = \operatorname{diag}\!\left(\tfrac{1}{d_1}, \dots, \tfrac{1}{d_n}\right)$
Orthogonal matrices
Orthogonality
$AA^T = I \;\Longrightarrow\; A^{-1} = A^T,\quad \det A = \pm 1$
Rotation matrices
Rotation composition
$R(\theta)\,R(\phi) = R(\theta + \phi), \qquad R(\theta)^n = R(n\theta)$

Reference tables (1)

The special-matrix catalog6 rows

Type	Defining property	Key consequence
Symmetric	$A^T = A$	$a_{ij} = a_{ji}$ ; inverse (if any) is symmetric
Skew-symmetric	$A^T = -A$	diagonal all 0; odd order $\Rightarrow \det = 0$ Odd-order skew-symmetric is ALWAYS singular (det 0). Even-order need not be.
Diagonal	off-diagonal all 0	$\det =$ product of diagonal entries
Orthogonal	$AA^T = I$	$A^{-1} = A^T$ ; $\det = \pm 1$
Idempotent	$A^2 = A$	$\det \in \{0, 1\}$
Involutory	$A^2 = I$	$A^{-1} = A$ ; $\det = \pm 1$

Recognise the defining equation first; the determinant and inverse follow immediately.

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

Consider the following statements: I. If

n\times n

(

n>1

) matrix is symmetric, then its inverse is also a symmetric matrix. II. If

n\times n

(

n>1

) matrix is singular, then its adjoint is also a singular matrix. Which of the statements given above is/are correct?

[Q18 · Apr · 2026]

Example 2Matrices & DeterminantsMODERATE

Consider the following for the items that follow: Let

A = \begin{pmatrix} 0 & \sin^2\theta & \cos^2\theta \\ \cos^2\theta & 0 & \sin^2\theta \\ \sin^2\theta & \cos^2\theta & 0 \end{pmatrix}

and

A = P + Q

where P is symmetric matrix and Q is skew-symmetric matrix.

What is Q equal to?

[Q38 · Sep · 2022]

Example 3Matrices & DeterminantsMODERATE

If matrix

A = \begin{pmatrix} 1-i & i \\ -i & 1-i \end{pmatrix}

where

i = \sqrt{-1}

, then which one of the following is correct?

[Q1 · Apr · 2020]

Example 4Matrices & DeterminantsMODERATE

A=\begin{pmatrix}\sin2\theta & -\cos2\theta & 0\\\cos2\theta & \sin2\theta & 0\\0 & 0 & 1\end{pmatrix}

, then which of the following statements is/are correct? (A)

A^{-1}=\text{adj }A

(B)

A

is skew-symmetric matrix. (C)

A^{-1}=A^T

Select the correct answer using the code given below:

[Q29 · Apr · 2024]

Example 5Matrices & DeterminantsHARD

Consider the following statements in respect of a nonsingular matrix

A

of order

n

: A.

A\,\text{adj}(A^T)=A(\text{adj }A)^T

B. If

A^2=A

, then

A

is identity matrix of order

n

. C. If

A^3=A

, then

A

is identity matrix of order

n

. Which of the statements given above are correct?

[Q12 · Apr · 2024]

Drill every past-year question on this subtopic

22 questions from the bank — paginated, with cart and Word-export support.

Drill the 22 questions