GATE Overflow Test Series | Mock GATE | Test 4 | Question: 36

Question

Consider the system of equations given below:

• $2x_1 + 4x_2 + x_3 + 2.6x_4 = 0$
• $2.6x_1 + 4.6x_2 + 1.6x_3 + 3.2x_4 = 0$
• $x_1 + 2x_2 + 3x_3 + 3.6x_4 = 0$
• $3.2x_1 + 5.2x_2 + 2.2x_3 + 3.8x_4 = 0$

The number of independent solutions of the system of equations is?

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

$=\begin{bmatrix} 2 & 4 & 1 & 2.6 \\ 2.6 & 4.6 & 1.6 & 3.2 \\ 1 & 2 & 3 & 3.6 \\ 3.2 & 5.2 & 2.2 & 3.8 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} =0$

$R_1 = R_1 \times 5$, $R_2=R_2 \times 5$, $R_3 = R_3 \times 5$, $R_4 = R_4 \times 5$

$= \begin{bmatrix} 10 & 20 & 5 & 13 \\ 13 & 23 & 8 & 16 \\ 5 & 10 & 15 & 18 \\ 16 & 26 & 11 & 19 \end{bmatrix}$

$R_4 = R_4-R_2$, $R_2=R_2-R_1$

$=\begin{bmatrix} 10 & 20 & 5 & 13 \\ 3 & 3 & 3 & 3 \\ 5 & 10 & 15 & 18 \\ 3 & 3 & 3 & 3 \end{bmatrix}$

$R_4=R_4-R_2$

$=\begin{bmatrix} 10 & 20 & 5 & 13 \\ 3 & 3 & 3 & 3 \\ 5 & 10 & 15 & 18 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

Rank $=3$

Number of independent solutions $=$ Nullity $=$ Number of unknowns(variables) - Rank

$=4-3$

$=1$.

Comments

Sir, how is entry (2,1) 13?

The rank of the matrix is 4. Therefore the determinant is not 0. Since it is a homogeneous system, and determinant is non-zero the system has a trivial solution(unique).

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2.6*5=13