A root of the equation \(\begin{vmatrix} 3-x&-6 &3 \\ -6 & 3-x & 3 \\ 3 &3 &-6-x \end{vmatrix}=0 \ is\)

(a)

6

(b)

3

(c)

0

(d)

-6

The value of the determinant of A = \(\begin{bmatrix} 0&a &-b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}is\)

(a)

-2abc

(b)

abc

(c)

0

(d)

a² + b² + c²

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (x₁, y₁ ), (x₂, y₂), (x₃, y₃ ) are

(a)

vertices of an equilateral triangle

(b)

vertices of a right angled triangle

(c)

vertices of a right angled isosceles triangle

(d)

collinear

If a \(\neq\) b, b, c satisfy \(\begin{vmatrix} a&2b &2c \\3 & b & c \\ 4 & a & b \end{vmatrix}=0,\) then abc =

(a)

a + b + c

(b)

0

(c)

b³

(d)

ab + bc

If A = \(\begin{vmatrix}-1 & 2 &4 \\ 3 &1 &0 \\ -2& 4 &2 \end{vmatrix}\) and B = \(\begin{vmatrix}-2 & 4 &2 \\ 6 &2 &0 \\ -2& 4 &8 \end{vmatrix}\), then B is given by

(a)

B = 4A

(b)

B = -4A

(c)

B = -A

(d)

B = 6A

If \(\left\lfloor . \right\rfloor \) denotes the greatest integer less than or equal to the real number under consideration and −1\(\le\) x < 0, 0 \(\le\) y < 1, 1 \(\le\) z < 2, then the value of the determinant \(\begin{vmatrix} \left\lfloor x \right\rfloor +1& \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor +1& \left\lfloor z \right\rfloor \\ \left\lfloor x \right\rfloor & \left\lfloor y \right\rfloor & \left\lfloor z \right\rfloor +1\end{vmatrix}\) is

(a)

\(\left\lfloor z \right\rfloor \)

(b)

\(\left\lfloor y \right\rfloor \)

(c)

\(\left\lfloor x \right\rfloor \)

(d)

\(\left\lfloor x \right\rfloor \)+1

If A is skew-symmetric of order n and C is a column matrix of order n \(\times\) 1, then C^T AC is

(a)

an identity matrix of order n

(b)

an identity matrix of order 1

(c)

a zero matrix of order 1

(d)

an identity matrix of order 2

The matrix A satisfying the equation \(\begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}\) A = \(\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}\) is

(a)

\(\begin{bmatrix} 1 & 4 \\ -1 & 0 \end{bmatrix}\)

(b)

\(\begin{bmatrix} 1 & -4 \\ 1 & 0 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}\)

(d)

\(\begin{bmatrix} 1 & -4 \\ 1 & 1 \end{bmatrix}\)

If A + I =\(\begin{bmatrix} 3& -2 \\ 4 & 1 \end{bmatrix}\), then (A + I )(A - I) is equal to

(a)

\(\begin{bmatrix} -5& -4 \\ 8 & -9 \end{bmatrix}\)

(b)

\(\begin{bmatrix} -5& 4 \\ -8 & 9 \end{bmatrix}\)

(c)

\(\begin{bmatrix} 5& 4 \\ 8 & 9 \end{bmatrix}\)

(d)

\(\begin{bmatrix} -5& -4 \\ -8 & -9 \end{bmatrix}\)

Let A and B be two symmetric matrices of same order. Then which one of the following statement is not true?

(a)

A + B is a symmetric matrix

(b)

AB is a symmetric matrix

(c)

AB = (BA)^T

(d)

A^T B = AB^T