{ x+2 y+3 z =c -2 x+y =d 3 x-y+z =e . (a) [ c d e ]=[

{ x+2 y+3 z =c -2 x+y =d 3 x-y+z =e . ...

Key Concepts

Matrix Rank

The rank of a matrix is the maximum number of linearly independent rows or columns it contains. In the context of linear systems, comparing the rank of the coefficient matrix with the rank of the augmented matrix helps to determine if the system is consistent and whether the solution is unique or has free parameters.

Consistency and Solution Existence

Consistency refers to whether a system of linear equations has at least one solution. An inconsistent system has no solution, while a consistent system may have a unique solution or infinitely many solutions. Determining consistency typically involves analyzing the rank of the coefficient matrix in relation to the augmented matrix.

Matrix Representation

Matrix representation involves expressing a system of linear equations in the form Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the constants vector. This form is useful for applying systematic solution methods and for analyzing the structure of the system.

Augmented Matrix

An augmented matrix is formed by appending the constants vector to the coefficient matrix, representing the entire system in a compact form. It is used for techniques like Gaussian elimination to determine the solvability of the system and to find solutions.

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. These systems can have a unique solution, infinitely many solutions, or no solution at all. Understanding these systems involves exploring relationships between equations and determining the existence and uniqueness of solutions.

Transcript

00:01 In the cution given system of equations that are x plus y plus 3 z is equal to c and 2 minus 2x plus y is equal to d and 3x minus y plus z is equal to e now in the question it is asking to find the solution by using inverse method and in the question it have two parts so first part given data is given matrix cede is equal to that is minus 1 for 2 so first i will try to find the answer for part a so it means we have to find the solution of given equation so first step is we have to write the matrix equation that is a x is equal to b or i can say that after multiplying on both side with a inverse then i will get that is x is equal to a inverse b here a is a coefficient matrix which is equal to elements are in the bracket that is coefficient of x y z so it will be written as one one three then minus two one zero then three minus one and one and one and b is a coefficient so, b is a matrix of cde.

01:38 So given as for part a, that is minus 1, 4, 2 elements are, and matrix x is a matrix of x, y, now next step is we have to find the a inverse, means inverse of matrix a.

02:01 So after sorting, i will get the a inverse that will be equal to 1 divided by 2.

02:09 In the bracket the elements are that is 1 minus 5 minus 3 then 2 minus 8 minus 6 and in the third row elements are 1 7 5 so this is the value of a inverse and now put the this all data in equation matrix equation that is x is equal to a inverse b so it will be written as that is x means matrix of x y z will be equal to a inverse so a inverse is that is which i had written in previous step that is 1 divided by 2 and in the brackets the elements are that is 1 minus 5 minus 3 in the second row elements are minus sorry 2 minus 8 minus 6 and in the third row that is 1 7 5 now this is a matrix which is multiplying by matrix b so matrix b is that is matrix of 1 minus 1 4 2 so for next step we have as you see we have to multiply these two matrix by using product dot product so i will get the elements so it will return as on the left side that is matrix of x y z will be equal to and on the right side that is half and in the black at the first element will be after multiplying first row with the column, means this first row with this column with matching elements.

03:57 So i will, so i will get that is, it will be written as 1 times of minus 1, minus 5 times of 4 and minus 3 times of 2.

04:11 Now for second element, we have to multiply second row with the column.

04:17 So means this second row with this column so it will be written as that is two times of minus one plus sorry minus eight times of four and minus six times of two now for third element we have to multiply this third row with the column so i can say that it will be written as that is one times of minus 1 plus 7 times of 4 plus 5 times of 2 now next step is simplify this in the bracket so i will get the elements that is 1 divided by 2 and the first element will be that is after solving my that is minus 1 minus 20 minus 6 so it will give that is minus 27...

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