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Test: Linear Algebra- 1 - 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

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Test: Linear Algebra- 1 - Question 5

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

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Test: Linear Algebra- 1 - Question 6

If the matrix A is such that

then the determinant of is equal to ______.

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Test: Linear Algebra- 1 - Question 7

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

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Test: Linear Algebra- 1 - Question 8

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

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Test: Linear Algebra- 1 - Question 9

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

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Test: Linear Algebra- 1 - 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?

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Test: Linear Algebra- 1 - 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 ___________

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Test: Linear Algebra- 1 - 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?

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Test: Linear Algebra- 1 - 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?

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Test: Linear Algebra- 1 - 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?

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Test: Linear Algebra- 1 - Question 16

Consider the matrix as given below.

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

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Test: Linear Algebra- 1 - Question 17

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

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Test: Linear Algebra- 1 - Question 18

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

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Test: Linear Algebra- 1 - 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 _______

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