AIEEE 2008 | Matrices and Determinants Question 359 | Mathematics

Let $$A$$ be $$a\,2 \times 2$$ matrix with real entries. Let $$I$$ be the $$2 \times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal entries of $$a$$. Assume that $${a^2} = I.$$
Statement-1 : If $$A \ne I$$ and $$A \ne - I$$, then det$$(A)=-1$$
Statement- 2 : If $$A \ne I$$ and $$A \ne - I$$, then tr $$(A)$$ $$ \ne 0$$.

statement - 1 is false, statement -2 is true

statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1.

statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.

statement - 1 is true, statement - 2 is false.

AIEEE 2008

MCQ (Single Correct Answer)

-1

Out of Syllabus

Let $$A$$ be a square matrix all of whose entries are integers.
Then which one of the following is true?

If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are not necessarily integers

If det $$A \ne \pm 1,$$ then $${A^{ - 1}}$$ exists and all its entries are non integers

If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are integers

If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ need not exists

AIEEE 2007

MCQ (Single Correct Answer)

-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

$$1/5$$

$$5$$

$${5^2}$$

$$1$$

AIEEE 2007

MCQ (Single Correct Answer)

-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 :

divisible by $$x$$ but not $$y$$

divisible by $$y$$ but not $$x$$

divisible by neither $$x$$ nor $$y$$

divisible by both $$x$$ and $$y$$