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# Linear Algebra MCQ Level - 1

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

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This mock test of Linear Algebra MCQ Level - 1 for Physics helps you for every Physics entrance exam. This contains 10 Multiple Choice Questions for Physics Linear Algebra MCQ Level - 1 (mcq) to study with solutions a complete question bank. The solved questions answers in this Linear Algebra MCQ Level - 1 quiz give you a good mix of easy questions and tough questions. Physics students definitely take this Linear Algebra MCQ Level - 1 exercise for a better result in the exam. You can find other Linear Algebra MCQ Level - 1 extra questions, long questions & short questions for Physics on EduRev as well by searching above.
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 :

Solution:

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

The correct answer is: n – r

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

Solution:

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

QUESTION: 3

### If then A-1 is equal to :

Solution:  The correct answer is: 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 :

Solution: 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: QUESTION: 5

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

Solution:

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

QUESTION: 6

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

will have no solution, unless :

Solution:   Hence, the system won't contain any solution unless a + b + c becomes 0.

The correct answer is: a + b + c = 0

QUESTION: 7

Solving will give,

Solution:

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

QUESTION: 8 Solution:
QUESTION: 9

If 1 – i, 7 + i, i and 2 are the eigenvalues of same matrix A. Then what would be the eigenvalues of Aθ.

Solution:

We know that, if λ is an eigenvalue of A. Then will be the eigenvalue of  Aθ.

So, the conjugates of eigenvalues of A will give the eigenvalues for Aθ.

The correct answer is: 1 + i, 7 – i, –i, 2

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 :

Solution:

If P  and Q  be m × n   and  n × p  matrices respectively, then rank  PQ ≤ min (rank P, rank Q)

We have,
rank (P) = n
rank (Q) = p
rank (PQ) = min(n, p) = p