If the system of linear equations x + ay + z | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) For a system of linear equations to have no solution,the determinant of the coefficient matrix $\Delta$ must be zero,and at least one of the Crame

Vedclass · Accepted Answer

$a = -1, b 
eq 9$

Vedclass · Answer

(D) For a system of linear equations to have no solution,the determinant of the coefficient matrix $\Delta$ must be zero,and at least one of the Cramer's rule determinants $(\Delta_1, \Delta_2, \Delta_3)$ must be non-zero.First,calculate $\Delta$:$\Delta = \begin{vmatrix} 1 & a & 1 \ 1 & 2 & 2 \ 1 & 5 & 3 \end{vmatrix} = 1(6 - 10) - a(3 - 2) + 1(5 - 2) = -4 - a + 3 = -a - 1$.Setting $\Delta = 0$,we get $-a - 1 = 0$,which implies $a = -1$.Next,calculate $\Delta_2$ (replacing the second column with the constants $3, 6, b$):$\Delta_2 = \begin{vmatrix} 1 & 3 & 1 \ 1 & 6 & 2 \ 1 & b & 3 \end{vmatrix} = 1(18 - 2b) - 3(3 - 2) + 1(b - 6) = 18 - 2b - 3 + b - 6 = 9 - b$.For the system to have no solution,we require $\Delta_2 
eq 0$,so $9 - b 
eq 0$,which implies $b 
eq 9$.Thus,the condition is $a = -1$ and $b 
eq 9$.

If the system of linear equations $x + ay + z = 3$,$x + 2y + 2z = 6$,and $x + 5y + 3z = b$ has no solution,then:

Similar Questions

If the system of linear equations $x - 4y + 7z = g$,$3y - 5z = h$,and $-2x + 5y - 9z = k$ is consistent,then:

If the system of equations $x+y+z=6$,$2x+5y+\alpha z=\beta$,and $x+2y+3z=14$ has infinitely many solutions,then $\alpha+\beta$ is equal to.

If $x=\alpha, y=\beta, z=\gamma$ is the solution of the system of equations $2x+3y+z=-1$, $3x+y+z=4$, and $x-3y-2z=1$, then the value of $\beta$ is:

Let $A = \begin{bmatrix} 2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b \end{bmatrix}$. If $A^3 = 4A^2 - A - 21I$,where $I$ is the identity matrix of order $3 \times 3$,then $2a + 3b$ is equal to:

Solve the system of linear equations using the matrix method: $x-y+z=4$,$2x+y-3z=0$,and $x+y+z=2$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module