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Solving a system of linear equation in two variables

The system ...

Question

The system $A X = B$ of (n) equations in (n) unknowns has infinitely many solutions if:

A

d e t A \neq 0, (a d j A) (B) = 0

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B

| A | = 0, (a d j A) (B) = 0

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C

| A | = 0, (a d j A) (B) \neq 0

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D

None of these

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Solution

The correct option is B $| A | = 0, (a d j A) (B) = 0$
Let $A X = B$ represent the system of n equations in n unknowns
If $| A | = 0$ and $(a d j A) B = 0$ , then the system has infinitely many solution.

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Similar questions

Q. If the system of equations x + ay = 0, az + y = 0, ax + z = 0 has infinitely many solutions then a = ___________________.

Let system of linear equations

a_{1} x + b_{1} y + c_{1} z = d_{1} a_{2} x + b_{2} y + c_{2} z = d_{2}

a_{3} x + b_{3} y + c_{3} z = d_{3}

can be expressed in the form

A X = B . . . . (*)

A = ⎛ ⎜ ⎝ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ⎞ ⎟ ⎠

B = ⎛ ⎜ ⎝ \begin{matrix} d_{1} d_{2} d_{3} \end{matrix} ⎞ ⎟ ⎠

X = ⎛ ⎜ ⎝ \begin{matrix} x y z \end{matrix} ⎞ ⎟ ⎠

The above system of equations (*) is said to be consistent with unique solution if A is non singular & the values of the variables x, y, z can be determined by using the equation

X = A^{- 1} B

and if A is singular then system of equations are either consistent with infinitely many solutions or inconsistent with no solution accordingly

(a d j A) (B) = 0

(a d j A) (B) \neq 0

where (adj A) is the transpose of cofactor matrix of A Now Assume

A = ⎛ ⎜ ⎝ \begin{matrix} a & 1 & 0 1 & b & d 1 & b & c \end{matrix} ⎞ ⎟ ⎠

B = ⎛ ⎜ ⎝ \begin{matrix} a & 1 & 1 0 & d & c f & g & h \end{matrix} ⎞ ⎟ ⎠

P = ⎛ ⎜ ⎝ \begin{matrix} f g h \end{matrix} ⎞ ⎟ ⎠

Q = ⎛ ⎜ ⎝ \begin{matrix} a^{2} 00 \end{matrix} ⎞ ⎟ ⎠

X = ⎛ ⎜ ⎝ \begin{matrix} x y z \end{matrix} ⎞ ⎟ ⎠

A X = P

possesses consistency with infinitely many solutions if?

Q. How many of the following statement are true ?

S_{1}

B

is symmetric matrix then

A B A^{T}

S_{2}

A^{4}

is singular matrix then

A

is also singular

S_{3}

A B = O

| A |

is non zero then

B

must be a null matrix

S_{4}

| A | \neq 0

(a d j A) B \neq 0

then matrix equation

A X = B

has no solution

x + a y = 0, y + a z = 0, z + a x = 0

. The value of a for which the system of equation has infinitely many solutions is

Q. Assertion :If a+b+c=0 and

a^{2} + b^{2} + c^{2} \neq b c + c a + a b

, then the system of homogenous equations

a x + b y + c z = 0

b x + c y + a z = 0

c x + a y + b z = 0

has infinite number of solutions. Reason: If |A|=0, the system of equations AX=B has infinite number of solutions.

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