Linear Algebra MCQ Level - 1

# Linear Algebra MCQ Level - 1

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## 10 Questions MCQ Test Topic wise Tests for IIT JAM Physics | Linear Algebra MCQ Level - 1

Linear Algebra MCQ Level - 1 for IIT JAM 2023 is part of Topic wise Tests for IIT JAM Physics preparation. The Linear Algebra MCQ Level - 1 questions and answers have been prepared according to the IIT JAM exam syllabus.The Linear Algebra MCQ Level - 1 MCQs are made for IIT JAM 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Linear Algebra MCQ Level - 1 below.
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Linear Algebra MCQ Level - 1 - Question 1

### Let  A be a m × n matrix with row rank = r = column rank. The dimension of the space of solution of the system of linear equations AX = 0 is :

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 1

Given that rank  A = r
⇒ There would be r  linearly independent solutions
Dim (A) = dim – rank = n – r

The correct answer is: n – r

Linear Algebra MCQ Level - 1 - Question 2

### What would be the dimension for the general solution of the homogeneous system. x1 + 2x2 – 3x3 + 2x4 – 4x5 = 0 2x1 + 4x2 – 5x3 + x4 – 6x5 = 0 5x1 + 10x2 – 13x3 + 4x4 – 16x5 = 0

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 2

Consider the coefficient matrix,  The system in echelon form has three free variables,  x3x4x5
hence dim = 3

Linear Algebra MCQ Level - 1 - Question 3

### If then A-1 is equal to :

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 3  The correct answer is: Linear Algebra MCQ Level - 1 - Question 4

A matrix M has eigen values 1 and 4 with corresponding eigen vectors (1, –1)T  and  (2, 1)T, respectively. Then M  is :

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 4 We know that if λ is an eigenvalue of M, then X is the corresponding eigen vector then, a12 – a22 = –1       ...(2) 2a21 + a22 = 4         ...(4)
Solving (1), (2), (3), (4), we get
a11 = 3, a12 = 2, a21 = 1, a22 = 2 The correct answer is: Linear Algebra MCQ Level - 1 - Question 5

If rank of matrix A is 5 and nullity of A is 3, then A is of order :

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 5

Rank is given by the number of non-zero rows the echelon from of the matrix and nullity is given by the Number of zero rows.

⇒   By sylvester's law, order of the matrix will be = rank + nullity
= 5 + 3
= 8

Linear Algebra MCQ Level - 1 - Question 6

The three equations,
–2x + y + z = a
x – 2y + z = b
x + y – 2z = c

will have no solution, unless :

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 6   Hence, the system won't contain any solution unless a + b + c becomes 0.

The correct answer is: a + b + c = 0

Linear Algebra MCQ Level - 1 - Question 7

Solving will give,

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 7

Consider the coefficient matrix, say  A, i.e. = 1(6 + 1) + 1(3+2) + 1(1 – 4)

= 9 ≠ 0
Hence, rank  A = 3 = Number of unknowns.
∴ There will be only one solution of the given matrix equation and that is

x = y = z = 0.
The correct answer is: (0 0 0)T

Linear Algebra MCQ Level - 1 - Question 8

The matrix A is represented as . The transpose of the matrix of this matrix is represented as?

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 8

Given matrix is a 3×2 matrix and the transpose of the matrix is 3 × 2 matrix. The values of matrix are not changed but the elements are interchanged, as row elements of a given matrix to the column elements of the transpose matrix and vice versa but the polarities of the elements remains same.

Linear Algebra MCQ Level - 1 - Question 9

Find the values of x, y, z and w from the below condition. Detailed Solution for Linear Algebra MCQ Level - 1 - Question 9

5z = 10 + 5 => 5z = 15 => z = 3
5x = 2 + z => 5x = 5 => x = 1
5y = 3 + 7 => 5y = 10 => y = 2
5w = 2 + 2 + w => 4w = 4 => w = 1.

Linear Algebra MCQ Level - 1 - Question 10

Let P be a matrix of order m × n and Q be a matrix of order n × p, n ≠ p. If rank (P) = n and rank of (Q) = p, then rank (PQ)  is :

Detailed Solution for Linear Algebra MCQ Level - 1 - Question 10

For the matrix P (m * n), rank<= min{ m,n}

Similarly for matrix Q (n * p), rank <= min{n , p}

Now, the rank of PQ <= min {rank of P ,rank of Q}

=> rank (PQ) <= min{ min {m,n} ,min{n,p}}

=> rank (PQ) <= min {m, n, p}

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