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) $P=[-Rev -Imv]$

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

• A) Unit
• B) Nonzero symmetric
• C) Nonzero skew symmetric
• D) Zero

If n x n matrices A and B are similar, then they have the ________ eigenvalues (with the same multiplicities).

• A) additive inverse of each other
• B) same
• C) multiplicative inverse of each other
• D) distinct

Let ‘Ax = 0’ be a homogeneous linear system of ‘n’ equations and ‘n’ unknowns. Then, the coefficient matrix ‘A’ is invertible if and only if this system has ______ solution.

• A) trivial
• B) infinite many
• C) No
• D) non-trivial

Corresponding to highest Eigen value the eigenvector is $$\left( {\begin{array}{*{20}{c}} 1&6&1 \\\ 1&2&0 \\\ 0&0&3 \end{array}} \right)$$

• A) $$\left[ {\begin{array}{*{20}{c}} 0 \\\ 1 \\\ 2 \end{array}} \right]$$
• B) $$\left[ {\begin{array}{*{20}{c}} 2 \\\ 0 \\\ 1 \end{array}} \right]$$
• C) $$\left[ {\begin{array}{*{20}{c}} 2 \\\ 1 \\\ 0 \end{array}} \right]$$
• D) $$\left[ {\begin{array}{*{20}{c}} 1 \\\ 2 \\\ 0 \end{array}} \right]$$

If A is triangular, then det A is the product of the entries on the _________ of A.

• A) rows
• B) upper diagonal
• C) main diagonal
• D) columns

An $n \times n$ matrix is diagonalizable with n --------- eigenvalues

• A) 0
• B) Similar
• C) Distinct
• D) Identical