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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* linearl*y* 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.

*x*_{1} + 2*x*_{2} – 3*x*_{3} + 2*x*_{4} – 4*x*_{5} = 0

2*x*_{1} + 4*x*_{2} – 5*x*_{3} + *x*_{4} – *6*x* _{5}* = 0

5

Solution:

Consider the coefficient matrix,

The system in echelon form has three free variables, *x*_{3}, *x*_{4}, *x*_{5}

hence **dim = 3**

The correct answer is: 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

a_{12} – a_{22} = –1 ...(2)

2*a*_{21} + *a*_{22} = 4 ...(4)

Solving (1), (2), (3), (4), we get

*a*_{11} = 3, *a*_{12} = 2, *a*_{21} = 1, *a*_{22} = 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

The correct answer is: 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

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

Solution:

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.

QUESTION: 9

Find the values of x, y, z and w from the below condition.

Solution:

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.

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:

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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