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QUESTION: 1

Let A be an mxn matrix and B an n*m matrix.

It is given that determinant (I_{m + AB}) = determinant (I_{n + BA}) , where Ik is the k*k identity matrix. Using the above property, the determinant of the matrix given below is

Solution:

QUESTION: 2

The determinant of the matrix

Solution:

QUESTION: 3

Which one of the following does NOT equal

?

Solution:

QUESTION: 4

The determinant of the matrix

Solution:

As it's upper traingular matrix ... So determinant will be product of main diagonal element.

det(A) = 6*2*4* -1 = -48.

Similar concept can be appliead , if Matrix is lower triangular or Diagonal Matrix

QUESTION: 5

The matrix of coefficients either have no solution or have infinite solutions is system of equations are

Solution:

*Answer can only contain numeric values

QUESTION: 6

If the matrix A is such that

then the determinant of is equal to ______.

Solution:

Hi,

For this kind of matrices Determinant is zero.

A will be a 3x3 matrix where the first row will be 2 [1 9 5], second row will be -4 [1 9 5] and third will be 7 [1 9 5]. That is, all the rows of A are linearly dependent which means A is singular.

When matrix is singular |A| = 0

*Answer can only contain numeric values

QUESTION: 7

Two eigenvalues of a 3 x 3 real matrix P are (2 + √-1) and 3 . The determinant of P is _______

Solution:

The determinant of a real matrix can never be imaginary. So, if one eigen value is complex, the other eigen value has to be its conjugate.

So, the eigen values of the matrix will be 2+i, 2-i and 3.

Also, determinant is the product of all eigen values.

So, the required answer is (2+i)*(2-i)*(3) = (4-i^{2})*(3) = (5)*(3) = 15.

*Answer can only contain numeric values

QUESTION: 8

Suppose that the eigenvalues of matrix A are 1,2, 4. The determinant of (A^{-1})^{T} is _______________.

Solution:

QUESTION: 9

In the given matrix, one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are

Solution:

now consider each of the triplets as the value of x, y, z and put in these equations the one which satisfies is the answer.

why so because an eigen vector represents a vector which passes through all the points which can solve these equations.

so we can observe that only option B is satisfying the equations.

QUESTION: 10

Consider the following 2×2 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:

*Answer can only contain numeric values

QUESTION: 11

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4 - by - 4 symmetric positive definite matrix is ___________

Solution:

This is because eigen vectors corresponding to DIFFERENT eigen values of a REAL symmetric matrix are ORTHOGONAL to each other.

However, same eigen values they may not be.

And Dot -product of orthogonal vectors(perpendicular vectors ) is 0 (ZERO)

QUESTION: 12

real valued square symmetric matrix of rank Consider the following statements.

(I) One eigenvalue must be in

(II) The eigenvalue with the largest magnitude must be strictly greater than 5

Which of the above statements about eigenvalues of is/are necessarily CORRECT?

Solution:

Eigen values of Therefore second statement is false.

Since the rank of matrix is 2, therefore atleast one eigen value would be zero for n>3.

For n= 2, It can be proven that

Both λ_{1} and λ_{2} would be real because is a real symmetric matrix. Which implies that atleast one eigen value would be in

Now, to prove matrix, let us consider the matrix is is the eigen value

of this matrix.

(For real symmetric matrix, b=c and __<__ would be replaced by equal sign)

QUESTION: 13

Let A be the matrixWhat is the maximum value of x^{T} Ax where the maximum is taken over all x that are the unit eigenvectors of A?

Solution:

x = [x^{1} , x^{2}] be a unit eigen vector

i.e. x^{1} 2 + x^{2} 2 = 1

∴ { x is a unit Eigen vector}

x'Ax = x'Lx= Lx'x = L [x1,x2]' [x^{1},x^{2}] = L [x1^{2} + x2^{2}] = L(1) = L .

QUESTION: 14

Let A be a matrix with eigen values -5,-2,1,4. Which of the following is an eigen value of the matrix , where identity matrix?

Solution:

So, for our given matrix, we have

This is a

block matrix where the first and last and the second and third elements are the same. So, applying the formulafor determinant of a block matrix as given here (second last case)

Each of the eigen value of A is the solution of the equation . So, we can equate to any of the eigen value of A, and that will get our value of and that is one of the choice. For no other choice, this equation holds. So, (c) 2 is the answer.

*Answer can only contain numeric values

QUESTION: 15

Let be two matrices.

Then the rank of P + Q is ___________ .

Solution:

det (P + Q), So Rank cannot be 3, but there exists a submatrix such that determinant of submatrix is not 0.

QUESTION: 16

Consider the matrix as given below.

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

Solution:

The given matrix is a upper triangular matrix and the eigenvalues of upper or lower traingular matrix are the diagonal values itself. (Property)

QUESTION: 17

Which one of the following statements is TRUE about every matrix with only real eigenvalues?

Solution:

Trace is the sum of all diagonal elements of a square matrix.

Determinant of a matrix = Product of eigen values.

A) Is the right answer. To have the determinant negative ,atleast one eigen value has to be negative(but reverse may not be true). {you can take simple example with upper or lower triangular matrices. In the case option (b) , (c) and (d) reverse is always true .}

*Answer can only contain numeric values

QUESTION: 18

The product of the non-zero eigenvalues of the matrix is ____

Solution:

We can see that the rank of the given matrix is 2 (since 3 rows are same, and other 2 rows are also same). Sum of eigen values = sum of diagonals. So, we have two eigen values which sum to 5. This information can be used to get answer in between the following solution. Let Eigen value be X. Now, equating the determinant of the following to 0 gives us the values for X. To find X in the following matrix, we can equate the determinant to 0. For finding the determinant we can use row and column additions and make the matrix a triangular one. Then determinant will just be the product of the diagonals which should equate to 0.

Taking X out from R4, 2-X from R1, (so, X = 2 is one eigen value)

Now, we got a triangular matrix and determinant of a triangular matrix is product of the diagonal.

So (3-X) (-X) = 0 => X = 3 or X = 0. So, X = 3 is another eigen value and product of non-zero eigen values = 2 * 3 = 6.

*Answer can only contain numeric values

QUESTION: 19

If the characteristic polynomial of a (the set of real numbers) is and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is _______

Solution:

Given that λ = 2 is an eigen value. So, it must satisfy characterstic equation.

*Answer can only contain numeric values

QUESTION: 20

The larger of the two eigenvalues of the matrix

Solution:

For finding the Eigen Values of a Matrix we need to build the Characteristic equation which is of the form,

A - λI

Where A is the given Matrix.

λ is a constant

I is the identity matrix.

We'll have a Linear equation after solving A - λI. Which will give us 2 roots for λ.

6 is larger and hence is the Answer.

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