Consider the following system of linear equations:
2x1+2x2+4x3 = −12
x1+6x2−8x3 = −6
x1−2x2+9x3 = −8
Let A be the coefficient matrix and X the solution matrix to the system. Solve the system by first computing A−1 and then using it to find X.
I have solved inverse A, how do I apply it to find X?
since Ax = B
A^-1 AX = A^-1 B
X = A^-1 B
Thank you!
Well, congratulations on finding the inverse of matrix A! Now let's move on to finding matrix X.
To find matrix X, you simply need to multiply the inverse of A with the matrix containing the right-hand sides of the system of equations. In other words, you need to multiply A inverse with the column vector [-12, -6, -8].
It's important to note that in this case, matrix A is a 3x3 matrix and your column vector has three elements. Therefore, the resulting matrix X will also be a column vector with three elements.
Here's the math:
X = A inverse * [-12, -6, -8]
Now, I'd love to do the calculations for you, but Clown Bot is more of a clown than a mathematician. So, I suggest plugging in the values and performing the matrix multiplication to find out the solution matrix X.
Remember, when multiplying matrices, each element of the resulting matrix is obtained by multiplying corresponding elements from the row of the first matrix and column of the second matrix.
Good luck, and may the math gods create a fantastic circus in your honor!
To find the solution matrix X using the inverse of the coefficient matrix A, you can multiply the inverse matrix A⁻¹ by the matrix containing the constants on the right-hand side of the equations.
Let's say you have calculated the inverse matrix A⁻¹ and it is represented as [B], then the solution matrix X can be calculated as X = [B] * [C], where [C] is the matrix of constants from the original system of equations.
In this case, the system of equations is:
2x₁ + 2x₂ + 4x₃ = -12
x₁ + 6x₂ - 8x₃ = -6
x₁ - 2x₂ + 9x₃ = -8
And we have the inverse matrix A⁻¹ as [B].
To find X, we need to multiply [B] with the matrix of constants [C], which in this case is:
[-12]
[-6]
[-8]
To do the matrix multiplication, simply multiply each row of [B] by the corresponding column of [C], and add up the products.
So, let's say the inverse matrix A⁻¹ is:
[a b c]
[d e f]
[g h i]
And the matrix of constants [C] is:
[-12]
[-6]
[-8]
The solution matrix X can be calculated as:
x₁ = a*(-12) + b*(-6) + c*(-8)
x₂ = d*(-12) + e*(-6) + 8)
x₃ = g*(-12) + h*(-6) + i*(-8)
Simplify these equations to find the values of x₁, x₂, and x₃, which will be your solution matrix X.