Consider the linear system. { x+3 y+z =a^2 2 x+5 y+2 a z =0 x+y+a^2 z =-9 . For whi

Consider the linear system. { x+3 y+z =a^2 2 x+5 y+2 a...

Key Concepts

Linear Systems

A linear system is a collection of linear equations involving the same set of variables. Solving a linear system usually involves finding values for the variables that satisfy all equations simultaneously. This concept underpins many areas in mathematics and engineering, where finding unique, infinite, or no solutions is crucial to understanding the behavior of the system.

Consistency and Inconsistency

A system is consistent if it has at least one solution and inconsistent if it has none. Determining inconsistency often involves transforming the system (using methods like Gaussian elimination) to check for equations that reduce to contradictory statements, such as 0 = 1, which indicates that the system cannot be satisfied.

Matrix Rank and the Rank Theorem

The rank of a matrix is the number of linearly independent rows or columns. The Rank Theorem states that a system of linear equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix. This concept is crucial in testing for inconsistency because a difference in rank indicates that there's an extra constraint not satisfied by the potential solutions.

Parametric Dependence in Systems

When parameters appear in a system of linear equations, the existence and nature of solutions can depend on the values of these parameters. Analyzing how the parameter affects the coefficients or the augmented part of the matrix allows us to identify specific parameter values that lead to inconsistency by causing contradictions or degeneracies in the system.

Gaussian Elimination and Row Operations

Gaussian elimination is a systematic method for solving linear systems by using elementary row operations to reduce the system to a simpler form, such as row echelon form. This process helps reveal relationships between the equations and identify any inconsistencies, making it an essential tool for determining whether a system has a solution or is inconsistent.

Transcript

00:01 Hello students here we have been given a system of linear equations the first equation is x plus 3y plus z x plus 3y plus z is equal to a square similarly the second equation given here is 2x 2x plus 5y plus 5y plus 2 a z plus 2 a z plus 2 a z this is equal to 0 and the last equation given here is x plus y plus a square z a square z this is equal to minus 9 so we have given this system of equation in this we have to calculate the value of a such that the system is inconsistent so the given condition is that the system is inconsistent we have to calculate the value of a so for finding out the solution of the equation what we do first we just form the augmented matrix right so we try to calculate the solution of the equation and then we'll finally put the condition that it is solution its solution is inconsistent the system is coming out to be inconsistent so let's first of all form the augmented matrix so augmented matrix will be formed by the coefficients of x y z and constant in the given equation so for the first equation the coefficients are 1 3 1 a square right a matrix close the next row is 2 coefficient is 2 coefficient of y is 5 coefficient of z is 2 a it is 2 a so a so a so a so i'll write it properly to a and constant is zero last and final though the coefficients are 1 1 a square and minus 9 so this is our augmented matrix so while while solving the solution of the solution of the equation what we do first we will convert the diagonal element of the augmented matrix to 1 and the rest of the elements lying on the bottom of the diagonal element to 0 so in our next step we'll convert this element to 0 this element to 0 so from this for this what we will do here will this replace row 2 from row 2 to row 2 to row 1 so row 2 minus 2 row 1 and further for row 3 what we will do row 3 will subtract only row 1 from row 3 so row 3 minus row 1 so doing these steps what we will do here sorry what we will get here actually our first row will remain same because you are not changing it as such so 1 three leave some space here one three one and here we have a square so matrix close next row here we are what we are doing here r2 minus 2 so r2 here is 2 and r1 is 1 1 is 1 1 will be multiplied by minus 2 this will become minus 2 and r2 is already 2 so minus 2 so minus 2 plus 2 this will be 0 the next element here we have 5 so r2 is 5 r1 is 3 so 3 will be multiplied by minus 2 so this will become minus 6 so this will give us minus 1 next element here we have 2a 2a and r1 is 1 so 2a minus 2 this is 2a minus 2a minus 2 this is 2a minus 2 the next element in the constant here r2 is 0 and r1 is a square so this a square will be multiplied by minus 2 a square minus 2 a square so our final element will be minus 2a square now i will erase this this or just for rough work now in the next what we are doing in the next row we are simply subtracting r1 from the next row so 1 minus 1 1 1 1 1 1 0 2 2 2 element 1 minus 3 is 1 1 2 3 is a square minus 1 right here is r 3 is a square and r1 is 1 so a square minus 1 a square minus 1 and final element here 9 minus 9 and a square so minus 9 minus a square so i will write here minus 9 minus a square so if this is minus a square minus 9 minus a square so as you have discussed we have to convert we will we have to convert the diagonal element to minus sorry diagonal element to 1 so this is already minus and we have to convert this element to mine 1 this first element of diagonal is already 1 we have to convert the second element to 1 so for this we have to multiply this entire room with minus 1 right so in the next step what we will do here we'll multiply r2 with minus 1 so let's excuse me here i'll write it leaving some space because i'll do it calculation here so r2 will be multiplied by minus 1 this will become minus r2 so our new matrix will be our new matrix will be something like fun first row we are not changing to 1 3 1 and constant is a square a square matrix close next next row here we have zero so we are multiplying it in the minus 1 so this will be 0 itself minus 1m.

05:21 So this will be 0 itself minus 1 multiplied so so this will become 1.

05:25 Minus 1 will be multiplied here.

05:26 So this will become 2 minus 2.

05:29 2 .2.

05:32 And here what we will do, minus 1 is multiplied.

05:36 So this will become 2a square.

05:40 2a square.

05:41 And on the last row here we have 0, minus 2, a square minus 1.

05:49 And minus 9 minus a square.

05:54 So before moving ahead, i'll just copy this matrix to the next page.

05:58 Because we have no left space over there so our first row is 131 a square so 1 3 1 constant is a square matrix close coming to the next row here we have 0 1 2 minus 2a so 0 1 2a and the last element of this row is 2a 2a square 2a square the last row 6 0 minus 2 a square minus 1 0 minus 2 a square minus 1 i'll write it properly a square minus 1 and the last element here minus 9 minus a square minus a square now in the next step we will convert this element to 0 so for converting this what we will do we'll replace r3 with r3 plus 2 r1 so sorry plus 2 r2 because r2 here is 1 so this will be multiplied by 2 this will become 2 and r 3 is already minus 2 so 2 minus 2 will become 0 so for converting this element to 0 we have to perform the following operation so our matrix will look something like first and second row will remain same because we are not changing r1 and r2 so 1 3 and constant is a square a square on the other hand the second row here we have 0 1 2 minus 2 a and 2a square and in the third row we are subtracting we are calculating what we are doing here r3 plus 2 r2 so for first element here we have r3 and r2 both are 0 so this will be 0 as it is now coming to the second element here we have minus 2 so r3 is minus 2 r2 is 1 so r2 is 1 this 1 will be multiplied by 2 this will become 2 and 2 and minus 2 now coming to this element here we have r3 as a square minus 1 so a square minus 1 and here r2 is 2 minus 2 so this will be multiplied by 2 so so so this will come 4 plus 4 minus 4a...

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