1
AIEEE 2008
+4
-1
Out of Syllabus
Let $$A$$ be a square matrix all of whose entries are integers.
Then which one of the following is true?
A
If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are not necessarily integers
B
If det $$A \ne \pm 1,$$ then $${A^{ - 1}}$$ exists and all its entries are non integers
C
If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are integers
D
If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ need not exists
2
AIEEE 2007
+4
-1
If $$D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr } } \right|$$ for $$x \ne 0,y \ne 0,$$ then $$D$$ is :
A
divisible by $$x$$ but not $$y$$
B
divisible by $$y$$ but not $$x$$
C
divisible by neither $$x$$ nor $$y$$
D
divisible by both $$x$$ and $$y$$
3
AIEEE 2007
+4
-1
Out of Syllabus
Let $$A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|.$$ If $$\,\,\left| {{A^2}} \right| = 25,$$ then $$\,\left| \alpha \right|$$ equals
A
$$1/5$$
B
$$5$$
C
$${5^2}$$
D
$$1$$
4
AIEEE 2006
+4
-1
If $$A$$ and $$B$$ are square matrices of size $$n\, \times \,n$$ such that
$${A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),$$ then which of the following will be always true?
A
$$A=B$$
B
$$AB=BA$$
C
either of $$A$$ or $$B$$ is a zero matrix
D
either of $$A$$ or $$B$$ is identity matrix
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