1. Algebra — Matrices and Determinants

Types of matrices

Matrices are rectangular arrays of numbers or algebraic expressions categorized by their dimensions, symmetry, and structural properties.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Classification and Decomposition of Square Matrices

Any square matrix $A$ of order $n$ over $\mathbb{R}$ can be uniquely expressed as the sum of a symmetric matrix $S$ and a skew-symmetric matrix $W$, given by $$S = \frac{1}{2}(A + A^T) \quad \text{and} \quad W = \frac{1}{2}(A - A^T)$$

📐 Geometric Interpretation

A matrix acts as a linear transformation on $\mathbb{R}^n$. Symmetric matrices represent orthogonal coordinate scaling and reflections across principal axes, whereas skew-symmetric matrices induce pure rotations and infinitesimal shear deformations.

⚙️ Algebraic & Structural Meaning

Provides structural decomposition in vector spaces, simplifying eigenvalue problems, quadratic forms, and matrix exponentiations in linear algebra.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Square Matrix Requirement	`N/A`	Strictly Mandatory	Concepts like trace, determinant, inverse, symmetry, and eigenvalues are undefined or nonsensical for rectangular matrices $m \times n$ where $m \neq n$.
Field of Scalars ($\mathbb{R}$ or $\mathbb{C}$)	`N/A`	Strictly Mandatory	Definitions of Hermitian, skew-Hermitian, orthogonal, and unitary matrices depend critically on conjugation properties over the complex field $\mathbb{C}$ versus real field $\mathbb{R}$.
Invertibility Condition	`N/A`	Essential for specific matrix types	If $\det(A) = 0$, the matrix is singular, failing to possess an inverse or defining a degenerate projection.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Trace of Idempotent and Nilpotent Matrices

$$A^2 = A \implies \text{Tr}(A) = \text{rank}(A)$$

Rule: For an idempotent matrix, its trace equals its rank, and its eigenvalues are strictly $0$ or $1$.

Application: Frequently asked in UP PGT to quickly determine the rank or trace without full diagonalization.

Determinant of Skew-Symmetric Matrix

$$n \text{ is odd} \implies \det(A) = 0$$

Rule: Any real skew-symmetric matrix of odd order has a determinant identically equal to zero.

Application: Directly tested in UP TGT/PGT to instantly eliminate non-zero options for odd-order skew-symmetric determinants.

Nilpotent Matrix Index Property

$$A^k = O \implies \text{Eigenvalues of } A \text{ are all } 0$$

Rule: A nilpotent matrix of index $k$ has a characteristic polynomial $\lambda^n = 0$, making its trace and determinant both zero.

Application: Used in DSSSB PGT to test advanced matrix property identification within seconds.

Involuntary Matrix Property

$$A^2 = I \implies A^{-1} = A$$

Rule: An involuntary matrix is its own inverse; its eigenvalues can only be $+1$ or $-1$.

Orthogonal Matrix Transpose Property

$$A^T A = I \implies \det(A) = \pm 1$$

Rule: For an orthogonal matrix, the determinant is always $\pm 1$ and row/column vectors form an orthonormal set.

Rapid Elimination via Diagonal Elements

$$a_{ii} = 0 \quad \forall i \implies \text{Candidate for Skew-Symmetry}$$

Rule: If all principal diagonal entries are zero, check if $a_{ij} = -a_{ji}$ to confirm skew-symmetry instantly.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Confusing Symmetric with Skew-Symmetric Signs

Common Pitfall: Assuming $A^T = A$ implies $a_{ii} = 0$, or forgetting that diagonal elements of a skew-symmetric matrix must be zero.

Counterexample: Let $A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ ($A$ is symmetric, diagonal is non-zero).

Why it fails: Skew-symmetric matrices require $a_{ii} = -a_{ii} \implies a_{ii} = 0$, whereas symmetric matrices have no such restriction on diagonals.

🛡️ Exam Defense: Always check principal diagonal elements first: zeros indicate potential skew-symmetry, non-zeros permit symmetry.

⚠️ COMMON ERROR #02

Misinterpreting Orthogonal Matrix Definition

Common Pitfall: Confusing orthogonal matrices ($A^T A = I$) with symmetric matrices ($A^T = I$).

Counterexample: $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is both symmetric and orthogonal, but $A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ is orthogonal but not symmetric.

Why it fails: Orthogonal condition requires product with transpose to be identity, not equality to identity.

🛡️ Exam Defense: Always verify $A A^T = I$ and $A^T A = I$ independently.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Symmetric Matrix	$A$ is square	$A^T = A$	Invariance under reflection across principal diagonal	Entries satisfy $a_{ij} = a_{ji}$
Skew-Symmetric Matrix	$A$ is square	$A^T = -A$	Diagonal entries are $0$	Entries satisfy $a_{ij} = -a_{ji}$
Orthogonal Matrix	$A$ is real square	$A^T A = I$	Rigid rotation/reflection (preserves inner product)	Determinant is $\pm 1$
Idempotent Matrix	$A$ is square	$A^2 = A$	Projection operator onto a subspace	Repeated application yields no change

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $A$ is a square matrix such that $A^2 = I$, then what is the simplified value of $(A - I)^3 + (A + I)^3 - 7A$?

(A) $A$(B) $I$(C) $2A$(D) $3I$

📚 Method A: Formal Textbook Derivation

Expand binomial expressions: $(A-I)^3 = A^3 - 3A^2 + 3A - I$. Since $A^2 = I$, $A^3 = A$. Substituting gives $(A - 3I + 3A - I) + (A + 3I + 3A + I) - 7A = (4A - 4I) + (4A + 4I) - 7A = 8A - 7A = A$.

⚡ Method B: 25-Second Exam Shortcut

Substitute scalar $A = 1$ (since $A^2=1$): $(1-1)^3 + (1+1)^3 - 7(1) = 0 + 8 - 7 = 1$, corresponding to matrix $A$.

💡 Tip: Use scalar substitution for matrix polynomial identities when matrix commute rules hold.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If $A = \begin{pmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \end{pmatrix}$, then matrix $A$ is classified as:

(A) Symmetric matrix(B) Skew-symmetric matrix(C) Diagonal matrix(D) Scalar matrix

📚 Method A: Formal Textbook Derivation

Compute transpose $A^T$. The rows become columns: $A^T = \begin{pmatrix} 0 & -1 & 2 \\ 1 & 0 & -3 \\ -2 & 3 & 0 \end{pmatrix} = -A$. Since $A^T = -A$ and diagonal elements are zero, $A$ is skew-symmetric.

⚡ Method B: 25-Second Exam Shortcut

Observe diagonal elements are all zero and $a_{12}=1$, $a_{21}=-1$ (i.e., $a_{ij}=-a_{ji}$). Instantly mark Skew-symmetric.

💡 Tip: Check diagonal zero-property first for $3 \times 3$ matrices.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If $A$ and $B$ are symmetric matrices of the same order, then $AB - BA$ is a:

A Symmetric matrix

B Skew-symmetric matrix

C Diagonal matrix

D Zero matrix

Derivation & Analysis:

Let $P = AB - BA$. Then $P^T = (AB - BA)^T = B^T A^T - A^T B^T = BA - AB = -(AB - BA) = -P$. Thus, $P$ is skew-symmetric.

MCQ 2: The determinant of a skew-symmetric matrix of odd order is always:

A 1

B -1

C 0

D Undefined

Derivation & Analysis:

For any skew-symmetric matrix $A$ of order $n$, $\det(A^T) = \det(-A) = (-1)^n \det(A)$. When $n$ is odd, $\det(A) = -\det(A) \implies 2\det(A) = 0 \implies \det(A) = 0$.

MCQ 3: If $A$ is an orthogonal matrix, then $A^{-1}$ is equal to:

A $A$

B $A^T$

C $I$

D $-A$

Derivation & Analysis:

By definition of an orthogonal matrix, $A^T A = I$, which implies $A^{-1} = A^T$.

MCQ 4: If $A^2 = A$, then the matrix $(I + A)^3 - 7A$ is equal to:

A $A$

B $I$

C $2I$

D $3A$

Derivation & Analysis:

Using binomial expansion and $A^k = A$ for idempotent matrix, $(I+A)^3 = I + 3A + 3A^2 + A^3 = I + 3A + 3A + A = I + 7A$. Thus $(I+A)^3 - 7A = I + 7A - 7A = I$.

MCQ 5: Every square matrix can uniquely be expressed as the sum of a symmetric matrix and a:

A Diagonal matrix

B Skew-symmetric matrix

C Scalar matrix

D Unit matrix

Derivation & Analysis:

Standard theorem: $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$, where the first part is symmetric and the second is skew-symmetric.

MCQ 6: If $A$ is a skew-symmetric matrix, then $A^2$ is a:

A Symmetric matrix

B Skew-symmetric matrix

C Null matrix

D Identity matrix

Derivation & Analysis:

$(A^2)^T = (A^T)^2 = (-A)^2 = A^2$. Hence, $A^2$ is symmetric.

MCQ 7: If a matrix is both symmetric and skew-symmetric, then it must be a:

A Diagonal matrix

B Zero matrix

C Identity matrix

D Scalar matrix

Derivation & Analysis:

If $A = A^T$ and $A = -A$, then $A = -A \implies 2A = O \implies A = O$ (Zero matrix).

MCQ 8: If $A$ is a non-singular square matrix, then $\det(A^{-1})$ is equal to:

A $\det(A)$

B $-\det(A)$

C $(\det(A))^{-1}$

D $0$

Derivation & Analysis:

Since $A A^{-1} = I$, taking determinants gives $\det(A)\det(A^{-1}) = 1 \implies \det(A^{-1}) = \frac{1}{\det(A)} = (\det(A))^{-1}$.

MCQ 9: A square matrix $A$ for which $A^2 = O$ is called:

A Idempotent

B Nilpotent of index 2

C Involuntary

D Orthogonal

Derivation & Analysis:

A matrix $A$ satisfying $A^k = O$ for some positive integer $k$ is nilpotent; if $k=2$, it is nilpotent of index 2.

Q10

MCQ 10: If $A$ is a diagonal matrix of order $n$ with diagonal elements $d_1, d_2, \dots, d_n$, then $\det(A)$ is:

A $\sum d_i$

B $0$

C $d_1 d_2 \dots d_n$

D $1$

Derivation & Analysis:

The determinant of a diagonal matrix is the product of its principal diagonal entries: $\prod_{i=1}^n d_i$.

Common questions

By definition, $a_{ij} = -a_{ji}$. Setting $i = j$ gives $a_{ii} = -a_{ii}$, which implies $2a_{ii} = 0 \implies a_{ii} = 0$ for all real entries.

Confusing transpose properties with inverse properties, or failing to verify order compatibility when multiplying symmetric and skew-symmetric matrices.

Use scalar substitution (e.g., setting matrix equations to numerical values if commutative properties hold) or apply standard index reduction rules like $A^2 = I$ or $A^2 = A$.

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