The system of linear equations x + y + z = 2, | vedclass

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(D) The system of equations is given by:$x + y + z = 2$$2x + 3y + 2z = 5$$2x + 3y + (a^2 - 1)z = a + 1$First,calculate the determinant $D$ of the coef

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inconsistent when $|a| = \sqrt{3}$

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(D) The system of equations is given by:$x + y + z = 2$$2x + 3y + 2z = 5$$2x + 3y + (a^2 - 1)z = a + 1$First,calculate the determinant $D$ of the coefficient matrix:$D = \begin{vmatrix} 1 & 1 & 1 \ 2 & 3 & 2 \ 2 & 3 & a^2 - 1 \end{vmatrix}$Expanding along the first row:$D = 1(3(a^2 - 1) - 6) - 1(2(a^2 - 1) - 4) + 1(6 - 6)$$D = 3a^2 - 3 - 6 - 2a^2 + 2 + 4$$D = a^2 - 3$For a unique solution,$D 
eq 0$,so $a^2 - 3 
eq 0$,which means $|a| 
eq \sqrt{3}$.If $|a| = \sqrt{3}$,then $a^2 = 3$. The system becomes:$x + y + z = 2$$2x + 3y + 2z = 5$$2x + 3y + 2z = a + 1$Comparing the second and third equations,we have $5 = a + 1$,which implies $a = 4$. However,we assumed $|a| = \sqrt{3}$,so $a^2 = 3$. Since $4^2 
eq 3$,the system is inconsistent when $|a| = \sqrt{3}$ because the left-hand sides of the second and third equations are identical,but the constants are different ($5 
eq a + 1$ when $a^2 = 3$).Thus,the system is inconsistent when $|a| = \sqrt{3}$.

The system of linear equations $x + y + z = 2, 2x + 3y + 2z = 5$,and $2x + 3y + (a^2 - 1)z = a + 1$ is:

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