Can somebody show I how to use matrices to solve the following. I want to solve for Tab first. I wiill ask 9n other part later. It given Tad =1200.
.285Tab-.428Tac=0
-.428Tab-.285Tac+.384Tad=0
-0.857Tab-.857Tac+.923Tad=0
No need for matrices, since all these linear combinations just add to zero.
If AX = 0 (all matrices) then X=0
Or, using Cramer's rule, since all the numerators have a zero column, they are all zero.
I don't know what that is . I plug in Tad=1200 and I get
eq1) 285Tab-.428Tac=0
eq2) -.428Tab-.285Tac+460.8=0
eq3) -0.857Tab-.857Tac+1107.6=0
Now how do I use eq1 and eq2 to to solve for Tab and Tac?
To solve the system of equations using matrices, we can write the equations in matrix form and use matrix operations.
First, let's write the given equations in matrix form:
```
[ 0.285 -0.428 0 ] [ Tab ] [ 0 ]
[ -0.428 -0.285 0.384 ] [ Tac ] = [ 0 ]
[ -0.857 -0.857 0.923 ] [ Tad ] [ 0 ]
```
Now, we can represent this matrix equation as AX = B, where A is the coefficient matrix, X is the column matrix of the variables (Tab, Tac, and Tad), and B is the column matrix of the constants (0, 0, 0).
Let's define the matrices A, X, and B:
A = [ 0.285 -0.428 0 ]
[ -0.428 -0.285 0.384 ]
[ -0.857 -0.857 0.923 ]
X = [ Tab ]
[ Tac ]
[ Tad ]
B = [ 0 ]
[ 0 ]
[ 0 ]
To find the values of Tab, Tac, and Tad, we need to solve the equation AX = B for X.
To do this, we can use matrix inverses. The inverse of a matrix A is represented as A^(-1). The solution to the equation AX = B is given by X = A^(-1) * B.
To find A^(-1), we can use any method such as Gaussian elimination, matrix adjoint, or matrix inverse formula. Once we have A^(-1), we can multiply it with B to find X.
Here are the steps to solve the matrix equation:
1. Calculate the inverse of matrix A, denoted as A^(-1).
2. Multiply A^(-1) with B to find X: X = A^(-1) * B.
3. The resulting X will give you the values of Tab, Tac, and Tad.
Please note that calculating the inverse can be done using various methods. You can use an online matrix calculator or software such as MATLAB or Python's NumPy to find the inverse and solve the equation.