Question

The system of equations
$$\eqalign{ & \alpha x + y + z = \alpha - 1 \cr & x + \alpha y + z = \alpha - 1 \cr & x + y + \alpha z = \alpha - 1 \cr} $$
has infinite solutions, if $$\alpha $$ is

A. $$- 2$$  
B. either $$- 2$$ or 1
C. not $$- 2$$
D. 1
Answer :   $$- 2$$
Solution :
$$\eqalign{ & \alpha x + y + z = \alpha - 1 \cr & x + \alpha y + z = \alpha - 1; \cr & x + y + \alpha z = \alpha - 1 \cr} $$
\[\Delta = \left| \begin{array}{l} \alpha \,\,\,\,\,\,1\,\,\,\,\,\,\,1\\ \,1\,\,\,\,\,\,\alpha \,\,\,\,\,\,1\\ \,1\,\,\,\,\,\,\,1\,\,\,\,\,\,\alpha \end{array} \right|\]
$$\eqalign{ & = \alpha \left( {{\alpha ^2} - 1} \right) - 1\left( {\alpha - 1} \right) + 1\left( {1 - \alpha } \right) \cr & = \alpha \left( {\alpha - 1} \right)\left( {\alpha + 1} \right) - 1\left( {\alpha - 1} \right) - 1\left( {\alpha - 1} \right) \cr} $$
For infinite solutions, $$\Delta = 0$$
$$\eqalign{ & \Rightarrow \,\,\left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 1 - 1} \right] = 0 \cr & \Rightarrow \,\,\left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 2} \right] = 0 \cr & \Rightarrow \,\,\alpha = - 2,1; \cr & {\text{But }}\alpha \ne 1.\,\,\,\,\,\,\therefore \,\,\alpha = - 2 \cr} $$

Releted MCQ Question on
Algebra >> Matrices and Determinants

Releted Question 1

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$  be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$  be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

A. $$C$$ is empty
B. $$B$$  has as many elements as $$C$$
C. $$A = B \cup C$$
D. $$B$$  has twice as many elements as elements as $$C$$
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Releted Question 2

If $$\omega \left( { \ne 1} \right)$$  is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

A. 0
B. 1
C. $$i$$
D. $$\omega $$
View Answer
Releted Question 3

Let $$a, b, c$$  be the real numbers. Then following system of equations in $$x, y$$  and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$    $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$    $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$     has

A. no solution
B. unique solution
C. infinitely many solutions
D. finitely many solutions
View Answer
Releted Question 4

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

A. $$A + B = B + A$$
B. $$A + B = A - B$$
C. $$A - B = B - A$$
D. $$AB=BA$$
View Answer

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Matrices and Determinants

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