1
GATE CSE 2004
+2
-0.6
Let $$A$$ be and n$$\times$$n matrix of the folowing form.

What is the value of the determinant of $$A$$?

A
B
C
D
2
GATE CSE 2004
+2
-0.6
If matrix $$X = \left[ {\matrix{ a & 1 \cr { - {a^2} + a - 1} & {1 - a} \cr } } \right]$$
and $${X^2} - X + 1 = 0$$
($${\rm I}$$ is the identity matrix and $$O$$ is the zero matrix), then the inverse of $$X$$ is
A
$$\left[ {\matrix{ {1 - a} & { - 1} \cr {{a^2}} & a \cr } } \right]$$
B
$$\left[ {\matrix{ {1 - a} & { - 1} \cr {{a^2} - a + 1} & a \cr } } \right]$$
C
$$\left[ {\matrix{ { - a} & 1 \cr { - {a^2} + a - 1} & {a - 1} \cr } } \right]$$
D
$$\left[ {\matrix{ {{a^2} - a + 1} & a \cr 1 & {1 - a} \cr } } \right]$$
3
GATE CSE 2004
+2
-0.6
In an M$$\times$$N matrix such that all non-zero entries are covered in $$a$$ rows and $$b$$ columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is
A
$$\le a + b$$
B
$$\le \max \left\{ {a,\,b} \right\}$$
C
$$\le$$ $$\min \left\{ {M - a,\,N - b} \right\}$$
D
$$\le \min \left\{ {a,\,b} \right\}$$
4
GATE CSE 2003
+2
-0.6
Consider the following system of linear equations $$\left[ {\matrix{ 2 & 1 & { - 4} \cr 4 & 3 & { - 12} \cr 1 & 2 & { - 8} \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ \alpha \cr 5 \cr 7 \cr } } \right]$$\$

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of $$\alpha$$, does this system of equations have infinitely many solutions?

A
$$0$$
B
$$1$$
C
$$2$$
D
infinitely many
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