If A and B are n x n matrices, then det (AB) = ______________.

• A) det(A).B
• B) A.det(B)
• C) det(A).det(B)
• D) A.B

Each pair of eigenvalue and its corresponding eigenvector provides a solution of the equation $x’=Ax$ which is called ---------------- of the differential equation.

• A) Eigensolution
• B) Eigenfunction
• C)
• D)

What is the maximum possible number of pivots in a 4x6 matrix ?

• A) 6
• B) 8
• C) 4
• D) 10

An n x n matrix A is diagonalizable if and only if A has n linearly ________ eigenvectors.

• A) dependent
• B) independent
• C)
• D)

If $\lambda + 7$ is a factor of the characteristic polynomial of a matrix $C$ , then which of the following is the Eigenvalue of $C$ ?

• A) $\frac{1}{7}$
• B) 7
• C) 0
• D) -7

If x - 2 is a factor of the characteristic polynomial of matrix C then an eigenvalue of C is

• A) 1/2
• B) -2
• C) 2
• D) 0

A row replacement operation on A does not change the ____________.

• A) rows
• B) matrix
• C) columns
• D) determinant

A partitioned square matrix ‘A’ is said to be ______ matrix if the matrices on the main diagonal are square and all matrices above the main diagonal are zero.

• A) Block lower triangular
• B) Null
• C) Block upper triangular
• D) Diagonal-constant

$$An{\text{_____}} matrix~with~n~distinct~eigenvalues~is~diagonlizable$$

• A) $$n \times n$$
• B) None of these
• C) $$m \times n$$
• D) Null

For a real matrix A, complex Eigen values and associated Eigen vectors come in ---------

• A) Complex pairs
• B) similar pairs
• C) none of the above
• D) conjugate pairs

An n x n matrix with n distinct eigen values is _____.

• A) diagonalizable
• B) symmetric
• C) Hermitian
• D) invertible

A row interchange __________ the sign of the determinant.

• A) changes
• B) does not change
• C)
• D)

For a non-singular matrix $A$, $A^{-1}$ is equal to------

• A) $Adj(A)/det(A)$*
• B) $det(A)*Adj(A)$
• C) $det(A)*A$
• D) None of the above.

A matrix $[A]_{n \times n$ has both positive and negative eigenvalues so in this case origin behaves as a --------------

• A) Critical point
• B) Saddle point
• C)
• D)

If n x n matrices A and B are similar, then they have the __________ characteristic polynomial.

• A) same but opposite in sign
• B) inverse of the other
• C) same
• D) different

If $A$ is an invertible square matrix then

• A) ${\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^{-1}}$
• B) ${\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}$
• C) ${\left( {{A^T}} \right)^T} = {\left( {{A^{ - 1}}} \right)^T}$
• D) None of the above

Suppose that real solutions $y_1$ and $y_2$ of $x’=Ax$, form a basis for the two-dimensional real vector space if $y_1$ and $y_2$ are ……..

• A) Linearly Independent
• B) Linearly dependent
• C)
• D)

Let A be an n ×n matrix. The number x is an eigenvalue of A if there exists a non-zero vector v such that Av = _______

• A) x. Av
• B) x . A
• C) x . v
• D) A. xv

If one of the eigenvalues of $[A]_{n\times n}$ is zero, it implies --------

• A) The determinant of $[A]$ is zero
• B) The solution to $[A][X]=[C]$a system of equations is unique
• C) The solution to $[A][X]=[0]$ system of equations is trivial
• D) The determinant of $[A]$ is nonzero

Let $A$ be a real 2 by 2 matrix with complex eigen values $\lambda=a-b+{i}, (b\neq 0)$ and associated eigenvectors $v$ in $C^2$, then $A=PCP^{-1}$, where $P=--------------$

• A) $P=[-Rev Imv]$
• B) $P=[Rev Imv]$
• C) $P=[Rev -Imv]$
• D) $P=[-Rev -Imv]$