Question:
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Let $A = \begin{bmatrix} 3 & 0 & -1 \\ 3 & 0 & -1 \\ 4 & 0 & 5 \end{bmatrix}$, then $\text{Nullity}(A) = 1, \text{Rank}(A) = 2$.
Reason R : For a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$, $\text{Rank}(T) + \text{Nullity}(T) \neq n$.
In the light of the above statements, choose the correct answer from the options given below
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Let $A = \begin{bmatrix} 3 & 0 & -1 \\ 3 & 0 & -1 \\ 4 & 0 & 5 \end{bmatrix}$, then $\text{Nullity}(A) = 1, \text{Rank}(A) = 2$. Reason R : For a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$, $\text{Rank}(T) + \text{Nullity}(T) \neq n$. In the light of the above statements, choose the correct answer from the options given below
Assertion A : Let $A = \begin{bmatrix} 3 & 0 & -1 \\ 3 & 0 & -1 \\ 4 & 0 & 5 \end{bmatrix}$, then $\text{Nullity}(A) = 1, \text{Rank}(A) = 2$. Reason R : For a linear transformation $T: \mathbb{R}^n \to \mathbb{R}^n$, $\text{Rank}(T) + \text{Nullity}(T) \neq n$. In the light of the above statements, choose the correct answer from the options given below
Show Hint
Rank-Nullity Theorem is an equality: $\text{Rank}(T) + \text{Nullity}(T) = n$. Any assertion claiming $\neq n$ is immediately false!
Updated On: Jul 30, 2026
- Both A and R are true and R is the correct explanation of A
- Both A and R are true but R is NOT the correct explanation of A
- A is true but R is false
- A is false but R is true
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Concept
The Rank-Nullity Theorem states that for any linear transformation $T: V \to W$ where $V$ is finite-dimensional: \[ \text{Rank}(T) + \text{Nullity}(T) = \dim(V) \]
Step 2: Key Formulas and Approach
Find the rank of matrix $A$ using row reduction, then use the Rank-Nullity Theorem to calculate nullity. Compare with Assertion A and Reason R.
Step 3: Step-by-step Explanation
• Evaluating Assertion A: Given $A = \begin{bmatrix} 3 & 0 & -1
3 & 0 & -1
4 & 0 & 5 \end{bmatrix}$. Apply $R_2 \to R_2 - R_1$: \[ \begin{bmatrix} 3 & 0 & -1 0 & 0 & 0 4 & 0 & 5 \end{bmatrix} \] The rows $[3, 0, -1]$ and $[4, 0, 5]$ are clearly linearly independent. Therefore, the number of non-zero rows in row-echelon form is $2$, so $\text{Rank}(A) = 2$. By Rank-Nullity theorem for $3 \times 3$ matrix ($n = 3$): \[ \text{Nullity}(A) = 3 - \text{Rank}(A) = 3 - 2 = 1 \] Hence, Assertion A is true.
• Evaluating Reason R: Reason R states that $\text{Rank}(T) + \text{Nullity}(T) \neq n$. By the fundamental Rank-Nullity Theorem, $\text{Rank}(T) + \text{Nullity}(T) = n$ strictly. The claim that it is $\neq n$ contradicts the theorem. Hence, Reason R is false.
Step 4: Final Answer
Assertion A is true, but Reason R is false. Thus, Option (C) is correct.
The Rank-Nullity Theorem states that for any linear transformation $T: V \to W$ where $V$ is finite-dimensional: \[ \text{Rank}(T) + \text{Nullity}(T) = \dim(V) \]
Step 2: Key Formulas and Approach
Find the rank of matrix $A$ using row reduction, then use the Rank-Nullity Theorem to calculate nullity. Compare with Assertion A and Reason R.
Step 3: Step-by-step Explanation
• Evaluating Assertion A: Given $A = \begin{bmatrix} 3 & 0 & -1
3 & 0 & -1
4 & 0 & 5 \end{bmatrix}$. Apply $R_2 \to R_2 - R_1$: \[ \begin{bmatrix} 3 & 0 & -1 0 & 0 & 0 4 & 0 & 5 \end{bmatrix} \] The rows $[3, 0, -1]$ and $[4, 0, 5]$ are clearly linearly independent. Therefore, the number of non-zero rows in row-echelon form is $2$, so $\text{Rank}(A) = 2$. By Rank-Nullity theorem for $3 \times 3$ matrix ($n = 3$): \[ \text{Nullity}(A) = 3 - \text{Rank}(A) = 3 - 2 = 1 \] Hence, Assertion A is true.
• Evaluating Reason R: Reason R states that $\text{Rank}(T) + \text{Nullity}(T) \neq n$. By the fundamental Rank-Nullity Theorem, $\text{Rank}(T) + \text{Nullity}(T) = n$ strictly. The claim that it is $\neq n$ contradicts the theorem. Hence, Reason R is false.
Step 4: Final Answer
Assertion A is true, but Reason R is false. Thus, Option (C) is correct.
Was this answer helpful?
0
0
Top CUET PG Mathematics Questions
- If x, y, z in the axis of a given 3D Geometrical shape, then the given coordinate point (-4,7,-1) will lie in which octant
- See the adjacent figure and find the value of ∠A in the semicircle
- The figure (A) is showing which type of function?
- Find out the degree of the differential equation \(\frac {d^2t}{ds^2}+(\frac {dt}{ds})^2+2t=0\)
- Ram and Mohan can do a piece of work in 20 days. Mohan alone can do the work in 30 days. In how many days will Ram alone complete the work.
View More Questions
Top CUET PG Linear Algebra Questions
- Let V(F) be a finite dimensional vector space and T: V \(\to\) V be a linear transformation. Let R(T) denote the range of T and N(T) denote the null space of T. If rank(T) = rank(T\textsuperscript{2}), then which of the following are correct?
A. N(T) = R(T)
B. N(T) = N(T\textsuperscript{2})
C. N(T) \(\cap\) R(T) = \{0\
D. R(T) = R(T\textsuperscript{2})} - Let \(m, n \in \mathbb{N}\) such that \(m<n\) and \(P_{m \times n}(\mathbb{R})\) and \(Q_{n \times m}(\mathbb{R})\) are matrices over real numbers and let \(\rho(V)\) denotes the rank of the matrix V. Then, which of the following are NOT possible.
A. \( \rho(PQ) = n \)
B. \( \rho(QP) = m \)
C. \( \rho(PQ) = m \)
D. \( \rho(QP) = \lfloor(m+n)/2\rfloor \), where \(\lfloor \rfloor\) is the greatest integer function - Which of the following are subspaces of vector space \(\mathbb{R}^3\):
A. \( \{(x,y,z) : x+y=0\ \)
B. \( \{(x,y,z) : x-y=0\} \)
C. \( \{(x,y,z) : x+y=1\} \)
D. \( \{(x,y,z) : x-y=1\} \)} - If \( A = \begin{pmatrix} 2 & 4 & 1 \\ 0 & 2 & -1 \\ 0 & 0 & 1 \end{pmatrix} \) satisfies \( A^3 + \mu A^2 + \lambda A - 4I_3 = 0 \), then the respective values of \( \lambda \) and \( \mu \) are:
- Let A and B be two symmetric matrices of same order, then which of the following statement are correct:
A. AB is symmetric
B. A+B is symmetric
C. \( A^T B = AB^T \)
D. \( BA = (AB)^T \)
View More Questions
Top CUET PG Questions
- Given below are two statements:
Statement I: For Plato the meaning of the word 'Justice' is basically a conception in the mind only.
Statement II: For Plato the meaning of the word 'Justice' is fixed, mind-independently
In the light of the above statements, choose the correct answer from the options given below: - What is the electron count of OS6CO182-?
- XeF4 hydrolysis gives A + B, HF, and Oxygen. Find A and B
- What is more basic in pyrol and pyridine?
- Hydrolysis of Al2C3, Be2?
View More Questions