Question 1 :
The sum of the elements of the matrix {tex} U ^ { - 1 } {/tex} is

Question 2 :
If {tex}\mathrm A {/tex} and {tex}\mathrm B {/tex} are unimodular matrices, then adjoint of {tex}\mathrm { A P B }{/tex} is

Question 3 :
If the adjoint of a {tex} 3 \times 3 {/tex} matrix {tex} P {/tex} is {tex} \left[ \begin{array} { l l l } { 1 } & { 4 } & { 4 } \\ { 2 } & { 1 } & { 7 } \\ { 1 } & { 1 } & { 3 } \end{array} \right] , {/tex} then the<br>possible value(s) of the determinant of {tex} P {/tex} is (are)<br>

Question 4 :
<p>If <em>A</em> and <em>B</em> are symmetric matrices of the same order and</p> <p><em>X</em> = AB + BA and <em>Y</em> = AB − BA, then (XY)^<em>T</em> is equal to</p>

Question 6 :
If <em>A</em> and <em>B</em> are squares matrices such that <em>A</em>²⁰⁰⁶ = <em>O</em>and AB = <em>A</em> + <em>B</em>, then det(<em>B</em>) equals

Question 8 :
If {tex}\mathrm k \in \mathrm { C } {/tex} and both {tex}\mathrm A = \left(\mathrm a _ { i j } \right) _ { \mathrm n \times \mathrm n } {/tex} and {tex}\mathrm { k A} {/tex} are orthogonal matrices, then

Question 9 :
<p>If <em>A</em> and <em>B</em> are two square matrices such that <em>B</em> = − <em>A</em>^− 1 BA, then</p> <p>(<em>A</em>+<em>B</em>)² is equal to</p>

Question 10 :
If {tex} \mathrm A = \left[ \begin{array} { c c c } { 3 } & { - 3 } & { 4 } \\ { 2 } & { - 3 } & { 4 } \\ { 0 } & { - 1 } & { 1 } \end{array} \right] . {/tex} Then

Question 11 :
If both {tex}\mathrm { A - \frac { 1 } { 2 } I} {/tex} and {tex} \mathrm {A + \frac { 1 } { 2 } I} {/tex} are orthogonal matrices, then

Question 12 :
The number of values of {tex} k {/tex} for which the system of equations {tex} ( k + 1 ) x + 8 y = 4 k ; k x + ( k + 3 ) y = 3 k - 1 {/tex} has infinitely many solutions is

Question 13 :
Let {tex} A = \left[ \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right] {/tex} then

Question 14 :
The value of {tex} \left[ \begin{array} { l l l } { 3 } & { 2 } & { 0 } \end{array} \right] U \left[ \begin{array} { l } { 3 } \\ { 2 } \\ { 0 } \end{array} \right] {/tex} is

Question 15 :
Which of the following values of {tex} \alpha {/tex} satisfy the equation<br><br>{tex} \left| \begin{array} { c c c } { ( 1 + \alpha ) ^ { 2 } } & { ( 1 + 2 \alpha ) ^ { 2 } } & { ( 1 + 3 \alpha ) ^ { 2 } } \\ { ( 2 + \alpha ) ^ { 2 } } & { ( 2 + 2 \alpha ) ^ { 2 } } & { ( 2 + 3 \alpha ) ^ { 2 } } \\ { ( 3 + \alpha ) ^ { 2 } } & { ( 3 + 2 \alpha ) ^ { 2 } } & { ( 3 + 3 \alpha ) ^ { 2 } } \end{array} \right| = - 648 \alpha ? {/tex}<br>

Question 16 :
If the system of equations {tex} x + a y = 0 , a z + y = 0 {/tex} and {tex} a x + z = 0 {/tex} has infinite solutions, then the value of {tex} a {/tex} is

Question 17 :
{tex} A = \left[ \begin{array} { c c c } { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 1 } \\ { 0 } & { - 2 } & { 4 } \end{array} \right] {/tex} and {tex} I = \left[ \begin{array} { c c c } { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right] {/tex} and<br>{tex} A ^ { - 1 } = \left[ \frac { 1 } { 6 } \left( A ^ { 2 } + c A + d I \right) \right] , {/tex} then the value of {tex} c {/tex} and {tex} d {/tex} are<br>

Question 18 :
If {tex} \omega \neq 1 {/tex} is a cube root of unity and {tex} x + y + z = a ,\ x {/tex} {tex} + \omega y + \omega ^ { 2 } z = b ,\ x + \omega ^ { 2 } y + \omega z = c , {/tex} then

Question 19 :
Let {tex} P = \left[ a _ { i j } \right] {/tex} be a {tex} 3 \times 3 {/tex} matrix and let {tex} Q = \left[ b _ { i j } \right] , {/tex} where {tex} b _ { i j } = 2 ^ { i + j } a _ { i j } {/tex} for {tex} 1 \leq i , j \leq 3 . {/tex} If the determinant of {tex} P {/tex} is {tex} 2 , {/tex} then the determinant of the matrix {tex} Q {/tex} is

Question 20 :
If {tex} a , b {/tex} and {tex} c {/tex} are the sides of a triangle and {tex} A , B {/tex} and {tex} C {/tex} are the angles opposite to {tex} a , b {/tex} and {tex} c {/tex} respectively, then<br>{tex} \Delta = \left| \begin{array} { c c c } a ^ { 2 } & b \sin A & c \sin A \\ b \sin A & 1 & \cos A \\ c \sin A & \cos A & 1 \end{array} \right| {/tex} is independent of<br>