Orthogonal, Projection and Special Matrices Practice Questions for GATE DA: 5+ Solved Questions with Step-by-Step Solutions

Solve 5+ Orthogonal, Projection and Special Matrices practice questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Orthogonal, Projection and Special Matrices

Chapter Roadmap

Welcome to the geometry of linear transformations. This chapter bridges the gap between abstract matrix algebra and visual geometric intuition.

Your Learning Journey

1. Projection Matrices and Quadratic Forms

Classifying surfaces, Rayleigh quotient, centering matrix. High Weightage

2. Orthogonal and Involutory Matrices

Length-preserving transformations, rotations, own inverses. Moderate Weightage

By the end of this chapter, you will:

Instantly classify the shape of any quadratic form using eigenvalues.
Recognize projection matrices and deduce their eigenvalues without calculation.
Solve constrained optimization problems in seconds.

The Geometry of Quadratic Forms

A quadratic form is a scalar-valued polynomial where every term has a degree of exactly two.

For a column vector $x \in R^{n}$ and an $n \times n$ matrix $A$ , the quadratic form is written as:

Q (x) = x^{T} A x

Geometric Intuition

If you set $Q (x) = c$ for some constant $c$ , you get a geometric surface:

In 2D: Conic sections (ellipses, hyperbolas, parabolas).
In 3D: Quadric surfaces (ellipsoids, hyperboloids).

The matrix $A$ acts as the "DNA" of this surface, dictating its orientation, stretching, and curvature.

Orthogonal, Projection and Special Matrices: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Linear Algebra NAT

For the centering matrix $M = I_{n} - \frac{1}{n} 1 1^{T}$ , what is the value of $M 1$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct substitution question testing the action of the centering matrix on the all-ones vector.

Step 1: Write down the expression for $M 1$ :

$M 1 = (I_{n} - \frac{1}{n} 1 1^{T}) 1$

Step 2: Distribute the vector $1$ :

$M 1 = I_{n} 1 - \frac{1}{n} 1 (1^{T} 1)$

Step 3: Simplify the terms:

$I_{n} 1 = 1$ .
$1^{T} 1$ is the dot product of the all-ones vector with itself, which equals $n$ (the sum of $n$ ones).

Step 4: Substitute back:

$M 1 = 1 - \frac{1}{n} 1 (n) = 1 - 1 = 0$

Step 5: The result is the zero vector. In NAT format asking for "the value" or implying a magnitude/component context where 0 is the unique numeric answer, the answer is 0.

Answer: 0

Question 2 · Linear Algebra MCQ

If the eigenvalues of a symmetric matrix $A$ are $2$ and $- 3$ , the quadratic form $x^{T} A x$ is classified as:

A.
Positive Definite
B.
Negative Definite
C.
Indefinite
D.
Positive Semi-Definite

Correct Answer:

Step-by-Step Solution

Key idea: This is a classification question based on the signs of eigenvalues.

Step 1: Recall the classification rules for quadratic forms using eigenvalues ( $λ$ ):

Positive Definite: All $λ > 0$ .
Negative Definite: All $λ < 0$ .
Indefinite: Some $λ > 0$ and some $λ < 0$ .

Step 2: Examine the given eigenvalues: $2$ (positive) and $- 3$ (negative).

Step 3: Since there is a mix of positive and negative eigenvalues, the form is Indefinite.

Answer: C

Question 3 · Linear Algebra MCQ

Which of the following matrices represents the quadratic form $Q (x_{1}, x_{2}) = 3 x_{1}^{2} + 4 x_{1} x_{2} + 5 x_{2}^{2}$ ?

A.
$egin{pmatrix} 3 & 4 \ 0 & 5 nd{pmatrix}$
B.
$egin{pmatrix} 3 & 2 \ 2 & 5 nd{pmatrix}$
C.
$egin{pmatrix} 3 & 4 \ 4 & 5 nd{pmatrix}$
D.
$egin{pmatrix} 3 & 0 \ 4 & 5 nd{pmatrix}$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct application of the symmetric matrix representation of a quadratic form.

Step 1: Recall that for a quadratic form $Q (x) = x^{T} A x$ , the matrix $A$ must be symmetric.

Step 2: The diagonal entries $A_{ii}$ correspond to the coefficients of the squared terms $x_{i}^{2}$ . Here, $A_{11} = 3$ and $A_{22} = 5$ .

Step 3: The off-diagonal entries $A_{ij}$ (where $i \neq = j$ ) correspond to half the coefficient of the cross-term $x_{i} x_{j}$ . The coefficient of $x_{1} x_{2}$ is 4. Therefore, $A_{12} = A_{21} = 4/2 = 2$ .

Step 4: Construct the matrix $A = (3225)$ .

Answer: B

Question 4 · Linear Algebra MCQ

A square matrix $P$ is called a projection matrix if it satisfies which of the following conditions?

A.
$P^{2} = I$
B.
$P^{T} = P$
C.
$P^{2} = P$
D.
$P^{- 1} = P$

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition recall question regarding special matrices.

Step 1: Recall the definition of a projection matrix. A matrix $P$ is a projection if applying it twice is the same as applying it once. Geometrically, once a vector is projected onto a subspace, projecting it again doesn't change it.

Step 2: Mathematically, this property is written as $P (P x) = P x$ for all $x$ , which implies $P^{2} = P$ . This property is called idempotence.

Step 3: Check the options. Option C states $P^{2} = P$ .

Note: While orthogonal projections also satisfy $P^{T} = P$ , the general definition of a projection matrix only requires idempotence ( $P^{2} = P$ ).

Answer: C

Question 5 · Linear Algebra MCQ

The expression $Q (x) = x^{T} A x$ results in a:

A.
Vector
B.
Scalar
C.
Square Matrix
D.
Column Matrix

Correct Answer:

Step-by-Step Solution

Key idea: This is a fundamental definition question about the output type of a quadratic form.

Step 1: Analyze the dimensions. Let $x$ be an $n \times 1$ column vector. Then $x^{T}$ is a $1 \times n$ row vector. Let $A$ be an $n \times n$ matrix.

Step 2: Perform the multiplication steps:

$x^{T} A$ results in a $(1 \times n) \times (n \times n) = 1 \times n$ row vector.
$(x^{T} A) x$ results in a $(1 \times n) \times (n \times 1) = 1 \times 1$ matrix.

Step 3: A $1 \times 1$ matrix is effectively a single number, or a scalar.

Answer: B

A question from this chapter

Question 1

For the centering matrix M = I_{n} - \frac{1}{n} 1 1^{T}, what is the value of M 1 ?

Question 2

If the eigenvalues of a symmetric matrix A are 2 and - 3, the quadratic form x^{T} A x is classified as:

Question 3

Which of the following matrices represents the quadratic form Q (x_{1}, x_{2}) = 3 x_{1}^{2} + 4 x_{1} x_{2} + 5 x_{2}^{2} ?

Question 4

A square matrix P is called a projection matrix if it satisfies which of the following conditions?

Question 5

The expression Q (x) = x^{T} A x results in a:

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Orthogonal, Projection and Special Matrices Practice Questions for GATE DA: 5+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Orthogonal, Projection and Special Matrices

Chapter Roadmap

Your Learning Journey

1. Projection Matrices and Quadratic Forms

2. Orthogonal and Involutory Matrices

By the end of this chapter, you will:

The Geometry of Quadratic Forms

The Geometry of Quadratic Forms

Geometric Intuition

Orthogonal, Projection and Special Matrices: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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One place for everything

Honest progress

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