
How do you solve , and using matrices?
Answer
449.7k+ views
Hint: Use the matrix equation to represent the given system of equations. You need to solve this equation. Use to find and solve the equation after substituting the values.
Complete step by step solution:
The given system of equations has three unknown quantities and three linear equations. So, we can find the particular solution of the system, if it exists. Now, we will use matrices to solve this system.
The given system of linear equations is,
We can represent the system in the matrix form as .
Here, , and .
Now, we get from the matrix equation, if the matrix exists.
To confirm the existence of we check whether .
We have the formula to find given as, where is the element at the row and column of the matrix .
So, we get .
Since, , we conclude that exists.
Now we will use the formula to get . But firstly, we need to find .
The adjoint of matrix is the transpose of its cofactor matrix whose elements are the cofactors of each element in .
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of .
So, the cofactor matrix is .
Now adjoint is the transpose of the co-factor matrix. So, .
Now, using , we have .
So, using the equation we get
Hence, the solution of the , and is .
Note:
It is important to be careful while doing calculations in these questions. As, it would be tedious and time-consuming to re-check the calculations, once done. Also, remember that if the coefficient matrix doesn’t have an inverse, then the system either is inconsistent (does not have a solution) or has infinitely many solutions.
Complete step by step solution:
The given system of equations has three unknown quantities and three linear equations. So, we can find the particular solution of the system, if it exists. Now, we will use matrices to solve this system.
The given system of linear equations is,
We can represent the system in the matrix form as
Here,
Now, we get
To confirm the existence of
We have the formula to find
So, we get
Since,
Now we will use the formula
The adjoint of matrix
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
Cofactor of
So, the cofactor matrix is
Now adjoint is the transpose of the co-factor matrix. So,
Now, using
So, using the equation
Hence, the solution of the
Note:
It is important to be careful while doing calculations in these questions. As, it would be tedious and time-consuming to re-check the calculations, once done. Also, remember that if the coefficient matrix doesn’t have an inverse, then the system either is inconsistent (does not have a solution) or has infinitely many solutions.
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