Statement 1 The system of linear equations x+y+z=2,2...

Statement 1 The system of linear equations $x+y+z=2,2 x+y-z=3,3 x+2 y+k z=4$ has a unique solution if $k=0$ and Statement 2 The system of equations $a_{1} x+b_{1} y+c_{1} z=d_{1}, a_{2} x+b_{2} y+c_{2} z=d_{2}, a_{3} x+b_{3} y+c_{3} z=d_{3}$, has a unique solution if $\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right| \neq 0$

Statement 1
The system of linear equations $x+y+z=2,2 x+y-z=3,3 x+2 y+k z=4$ has a unique solution if $k=0$
and
Statement 2
The system of equations $a_{1} x+b_{1} y+c_{1} z=d_{1}, a_{2} x+b_{2} y+c_{2} z=d_{2}, a_{3} x+b_{3} y+c_{3} z=d_{3}$, has a unique solution if
$\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right| \neq 0$

Key Concepts

Determinant and Invertibility

The determinant of a square matrix is a scalar value that characterizes certain properties of the matrix, including its invertibility. For a system of linear equations, if the determinant of the coefficient matrix is nonzero, the matrix is invertible, which guarantees that the system has a unique solution. Conversely, a zero determinant indicates that the matrix is singular, and the system may have either no solutions or infinitely many solutions.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. These systems can have a unique solution, no solution, or infinitely many solutions, depending on the relationships among the equations. They are fundamental objects in linear algebra and are used to model many real-life problems in science and engineering.

Coefficient Matrix

The coefficient matrix is formed by the coefficients of the variables in a system of linear equations. It encapsulates the structure of the system and is essential for analyzing the system’s properties, such as solvability and the nature of its solutions.

Transcript

00:01 So in the given question we have two statements that have been given that have been given and the first statement the first statement what we are told is we have a system of linear equations given as x plus y plus z is equal to 2 2x plus y minus z is equal to 3 3 x plus plus 2y plus kz is equal to 4 and we are told that this system of equations has a unique solution as a unique solution if the value of k is equal to 0.

00:59 So this is given as the first statement.

01:03 And in the second statement what we have is we have the statement that the system of equations, the system of equations a1x plus b1y plus c1z as equal to 0, sorry, is equal to d1, a2x plus b2y plus c2z is equal to d2 and a3x a 3x plus c3 z is equal to d3 for this system of equations it would have a unique solution, has a unique solution unique solution if the determinant that is formed by the coefficients in this equation that is determinant of a1 b 1 c1 a 2 b 2 c2 a 3 a 3 is not equal to 0 so this is given us the second statement.

02:36 So what we should do over here is to check whether statement one and two are true or false, right? so what we can see over here is that in statement in statement two, that is saying that the system of equations have if they have a unique solution, the determinant is not to equal to 0 is in fact a property or a condition it is a standard condition of a system of equation right so what we can say over here is that statement 2 is true since it is a standard condition statement 2 is true right so what we need to check over here is the first statement right so in order to check the first statement let's take the coefficient matrix of the system of equations that have been given over here.

03:36 So it would be 1 -1 -21 -1 -3 -2 -k.

03:45 Right? so this is the required determinant and for this to have a unique solution, the determinant should be not equal to 0 and when we take the determinant what we would have is k plus 2 minus 2 times k plus 3 plus 4 minus 3 and this should not be equal to 0.

04:19 So what we would have is we would have k plus 2 minus 2 k minus 3, 4 minus 3 is 1.

04:35 Right so plus 1 is not equal to 0 so 2 plus 1 is equal to 3 3 minus 3 is equal to 0 and k minus 2k is equal to minus k right k minus 2k is equal to minus k from which we can just simplify and write it as k is not equal to right so what we have found over here is that when the when we take the determinant and if it has a unique solution the determinant should in fact not have a value that is equal to zero right so for the determinant to not have a value equal to zero the condition that we reached over here is that the value of k should not be equal to 0.

05:35 But in the statement 1 what we are told is that if the system has a unique solution, k has a value of 0...

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