If A is a m×n matrix and \[A = A^T\], which of the following must always be true?

• A) m = n
• B) m and n are different
• C)
• D)

The eigenvalues of a 4 by 4 matrix [A] are given as 2, -3, 1, 3 and 7. The |det(A)| then is

• A) 530
• B) 25
• C) 546
• D) cannot be determined

\[\left( {\begin{array}{*{20}{c}} 3&0&0 \\\ 5&4&0 \\\ 3&6&1 \end{array}} \right)\] The Eigen-values are

• A) (3,4,5)
• B) (3,4,1)
• C) (3,4,2)
• D) (2.3,5)

Let A be an n \times n matrix whose distinct eigen values are \lambda_1, \lambda_2, …, \lambda_p. The matrix A is diagonalizable if and only if the ----------- of the dimensions of the distinct eigen spaces is equal to n

• A) sum
• B) Product
• C)
• D)

\[\begin{gathered} {x_1} + 2{x_2} + 3{x_3} = 7 \hfill \\\\ 4{x_1} + {x_2} + 2{x_3} = 2 \hfill \\\\ - 4{x_1} + 3{x_2} + 9{x_3} = 4 \hfill \\\\\\ \end{gathered} \] The augmented matrix for the system is -----------

• A) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]$
• B) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}} \right]$
• C) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]$
• D) \[\left( {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}} \right)\]

If A is a m×n matrix and A = A^T, which of the following must always be true?

• A) m and n are different
• B) m = n
• C)
• D)

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

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

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

$\phi $ is the angle between the positive x-axis and the ray

• A) From (a,b) through (0,0)
• B) From (0,0) through (a,b)
• C) None of these
• D) From (1,1) through (0,0)

If A is m\cross n and b is in R^n, a least square solution of Ax=b is an \widehat x in R^n such that

• A) ||b-A\widehat x ||<=||b-Ax|| \forall x \in R^n
• B) ||b-A\widehat x||>=||b-Ax|| \forall x \in R^n
• C)
• D)

The zero vector is orthogonal to every vector in R^n

• A) True
• B) False
• C)
• D)

Let W is a finite dimensional subspace of an inner product space V and y is any vector in V. The best approximation to y from W is then

• A) Proj^{y}_{w}
• B) Proj^{w}_{y}
• C)
• D)

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

• A) Eigen functions
• B) Eigen vectors
• C)
• D)

A m ×n matrix U has orthonormal columns if and only if {U^t}U = I

• A) True
• B) False
• C)
• D)

Let W be a subspace of {R^n} and \{ {u_1},{u_2},...,{u_p}\} is an orthonormal basis, then

• A) \Pr o{j_w}y = (y.y){u_1} + (y.y){u_2} + ... + (y.y){u_p}
• B) \Pr o{j_w}y = (y.{u_1}){u_1} + (y.{u_2}){u_2} + ... + (y.{u_p}){u_p}
• C)
• D)

Let W be a subspace of {R^n} and \{ {u_1},{u_2},...,{u_p}\} is any orthogonal basis of W, then \mathop y\limits^\^ = {c_1}{u_1} + {c_2}{u_2} + ... + {c_n}{u_n} Where

• A) {c_j} = \frac{{y.{u_j}}}{{u_j}}
• B) None of these
• C) {c_j} = \frac{{y.{u_j}}}{y}
• D) {c_j} = \frac{{y.{u_j}}}{{{u_j}.{u_j}}}

If a vector x_0 is specified, then the initial value problem is to construct the unique function x such that --------

• A) x’=Ax and x(0)=x_0
• B) x’=Ax’ and x(0)=x_0
• C)
• D)

How many subspaces does R^2 have?

• A) None of these
• B) Infinitely many
• C) 0 and R^2
• D) 0, R×0,0×R,R^2

Let u, v and w be vectors in {R^n}, then

• A) (u+v).w=u.w+v.w
• B) (u+v).w=u.w+v.w-2u.w
• C) (u+v).w=u.w-v.w
• D) None of the above

Two vectors u and v in R^n are orthogonal if ------

• A) u.v \neq 0
• B) u.v=0
• C)
• D)