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QUESTION: 1

A square matrix is singular whenever:

Solution:

If the rows (or columns) of a square matrix are linearly dependent, then the determinant of matrix becomes zero.

Therefore, whenever the rows are linearly dependent, the matrix is singular.

QUESTION: 2

If A and B are real symmetric matrices of size n x n. Then, which one of the following is true?

Solution:

The matrix M is s aid to be sym m etric iff M^{T}= M

QUESTION: 3

Let a = (a_{ij}) be an n-rowed square matrix and I_{12} be the matrix obtained by interchanging the first and second rows of the n-rowed Identity matrix. Then AI_{12} is such that its first

Solution:

Let,

I_{12} is the matrix obtained by inter-changing the first and second row of the identity Matrix I.

So

AI_{12} is the matrix having first column same as the second column of A.

QUESTION: 4

The rank of the matrix given below is:

Solution:

The given matrix is:

QUESTION: 5

Consider the following determinant:

Which of the following is a factor of Δ?

Solution:

The determinant of a matrix can’t be affected by elementary row operation

So,

So (a - b) is a factor of Δ.

QUESTION: 6

The rank of the matrix is

Solution:

The given matrix is

Above matrix has only 1 independent row, so the given matrix has rank 1.

QUESTION: 7

The following system of equations:

Has a unique solution. The only possible value(s) for a is/are

Solution:

The augmented matrix for above system is

Now as long as a - 5 ≠ 0,

rank (A) = rank (A | B) = 3

∴ a can take any real value except 5.

QUESTION: 8

If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?

S_{1}: Each row of M can be represented as a linear combination of the other rows.

S_{2}: Each column of M can be represented as a linear combination of the other columns.

S_{3}: MX = O has a nontrivial solution.

S_{4}: M has an inverse.

Solution:

S_{1} and S_{2}:

Since M has zero determinant, its rank is not full i.e. if M is of size 3 x 3 , then its rank is not 3. So there is a linear combination of rows which evaluates to 0 i.e. k_{1}R_{1} + k_{2}R_{2 }+...+k_{n}R_{n} = 0 and there is a linear combination of columns which evaluates to 0 i.e.

Now any row R_{i} can be written as linear combination of other rows as:

Similar is the case for columns.

So S_{1} and S_{2} are true.

QUESTION: 9

Consider the matrix as given below:

Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?

Solution:

Since the given matrix is upper triangular, its eigen values are the diagonal elements themselves, which are 1, 4 and 3.

QUESTION: 10

Consider the following 2 x 2 matrix A where two elements are unknown and are marked by ‘a’ and ‘b'. The eigenvalues of this matrix are -1 and 7. What are the values of ‘a’ and 'b'?

Solution:

Trace = Sum of eigen values

1 + a = 6

⇒ a = 5

Determinant = Product of eigen values

⇒ b = 3

∴ a = 5, b = 3

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