Engineering Mathematics
Linear Algebra
Practice questions from Linear Algebra.
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IncorrectConsider the matrix below:
For which of the following combinations of , and , is the rank of at least three?
(i) and .
(ii) .
(iii) and .
(iv) .
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Sign in to UnlockConsider the vectors
For real-valued scalar variable , the value of
is _________ (rounded off to two decimal places).
denotes the Euclidean norm, i.e., for .
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Sign in to UnlockLet and denote the set of real numbers and the three-dimensional vector space over it, respectively. The value of for which the set of vectors
Dose not form a basis of is__________.
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Sign in to UnlockConsider the matrix where k is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?
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Sign in to UnlockLet and be two vectors. The value of the coefficient in the expression , which minimizes the length of the error vector , is
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Sign in to UnlockLet the sets of eigenvalues and eigenvectors of a matrix be and , respectively. For any invertible matrix , the sets of eigenvalues and eigenvectors of the matrix , where , respectively, are
and
and
and
and
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Sign in to UnlockLet be an real column vector with length . The trace of the matrix is
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Sign in to UnlockConsider a system of linear equations , where
This system of equations admits
a unique solution for
infinitely many solutions for
no solutions for
exactly two solutions for
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Sign in to UnlockLet be two non-zero real numbers and be two non-zero real vectors of size . Suppose that and satisfy , and . Let be the matrix given by:
The eigen values of are
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Sign in to UnlockIf the vectors and and are linearly dependent, the value of is ________.
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Sign in to UnlockA real non-singular matrix with repeated eigen value is given as
where is a real positive number. The value of (rounded off to one decimal place) is ________.
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Sign in to UnlockIf are six vectors in , which one of the following statements is False?
If spans , then it forms a basis for
These vectors are not linearly independent
It is not necessary that these vectors span .
Any four of these vectors form a basis for ,
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Sign in to UnlockConsider the following system of linear equation,
Which one of the following conditions ensures that a solution exists for the above system?
and
and
and
and
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Sign in to UnlockThe number of distinct Eigen values of the matrix
Is equal to _______
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Sign in to UnlockConsider matrix and vector. The number of distinct real values r of k for which the equation Ax=0 has infinitely many solutions is
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Sign in to UnlockLet M be a real matrix. Consider the following statements:
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct Eigen-values.
S3: M is non-singular (invertible). Which one among the following is TRUE?
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Sign in to UnlockConsider the following statements about the linear dependence of the real valued functions and , over the field of real numbers.
- and are linearly independent on
- and are linearly dependent on
- and are linearly independent on
- and are linearly dependent on
Which one among the following is correct?
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Sign in to UnlockConsider the 5 × 5 matrix
It is given that A has only one real eigenvalue. Then the real eigenvalue of A is
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Sign in to UnlockThe rank of the matrix
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Sign in to UnlockThe rank of the matrix is _______________.
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Sign in to UnlockLet , (where I denotes the identity matrix) and M ≠ I, and . Then, for any natural number k, equals:
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Sign in to UnlockThe value of x for which the matrix has zero as an eigenvalue is _____________.
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Sign in to UnlockThe matrix has det(A) = 100 and trace(A) = 14. The value of is __________.
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Sign in to UnlockConsider a 2 × 2 square matrix. , Where x is unknown. If the eigenvalues of the matrix A are and , then x is equal to
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Sign in to UnlockIf the vectors and form an orthogonal basis of the three-dimensional real space , then the vector u= (4, 3, -3) can be expressed as
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Sign in to UnlockConsider a system of linear equations:
The value of k for which the system has infinitely many solutions is ______.
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Sign in to UnlockThe value of p such that the vector is an eigenvector of the matrix is _______.
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Sign in to UnlockThe value of x for which all the Eigen-values of the matrix given below are real is
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Sign in to UnlockFor , the determinant of is
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Sign in to UnlockFor matrices of same dimension M, N and scalar c, which of these properties DOES NOT ALWAYS HOLD?
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Sign in to UnlockA real (4 x 4) matrix A satisfies the equation , where I is (4 X 4) identity matrix. The positive Eigen value of A is
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Sign in to UnlockConsider the matrix which is obtained by reversing the order of the columns of the identity matrix . Let , where ∝ is a non-negative real number. The value of ∝ for which det(P)=0 is _____________.
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Sign in to UnlockThe determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB is ________.
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Sign in to UnlockThe system of linear equations has
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Sign in to UnlockThe maximum value of the determinant among all 2 x 2 real symmetric matrices with trace 14 is __________.
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Sign in to UnlockWhich one of the following statements is NOT true for a square matrix A?
If A is upper triangular, the Eigen values of A are the diagonal elements of it
If A is real symmetric, the Eigen values of A are always real and positive
If A is real, the Eigen values of A and are always the same
If all the principal minors of A are positive, all the Eigen values of A are also positive
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Sign in to UnlockThe minimum Eigen value of the following matrix is
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Sign in to UnlockLet A be an matrix and B an matrix. It is given that determinant =determinant , where is the identity matrix. Using the above property, the determinant of the matrix given below is
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Sign in to UnlockGiven that
and , the value of is
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Sign in to UnlockThe system of equations
Has no solution for values of and given by
,
,
,
,
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Sign in to UnlockThe Eigen values of a skew-symmetric matrix are
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Sign in to UnlockThe Eigen values of the following matrix are
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Sign in to UnlockAll the four entries of the matrix are nonzero, and one of its eigenvalues is zero. Which of the following statements is true?
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Sign in to UnlockThe system of linear equations
Has
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Sign in to UnlockIt is given that are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors is
2M
M +1
M
Dependent on the choice of
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Sign in to UnlockThe rank of the matrix is
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Sign in to UnlockThe eigenvalues and the corresponding eigenvectors of a 2 × 2 matrix are given by
Eigenvalue | Eigenvector |
The matrix is:
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Sign in to UnlockFor the matrix the Eigen value corresponding to the eigenvector is:
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Sign in to UnlockGiven the matrix , the eigen vector is
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Sign in to UnlockLet and .
Then (a + b) =
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Sign in to UnlockGiven an orthogonal matrix, is
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Sign in to UnlockThe eigen values of the matrix are
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Sign in to UnlockThe eigen values o are
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Sign in to UnlockThe rank of an matrix cannot be more than m. (True=1,False=0)
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Sign in to UnlockIf . The matrix , calculated by the use of Cayley Hamilton theorem or otherwise, is _________.
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