Solve the matrix equation AX =B for X using:
A= [ 2 -3] B= [4 ]
[ 1 5] [2 ]
(theres a blank space between the 4 and the 2)
I found the inverse of A to be
5/13 3/13
-1/13 2/13
so
A-1 A X = A-1</sup B
X =
│ 5/13 3/13││4│
│-1/13 2/13││2│
=
│2│
│0│
that doesn't look like I intended.
let's try that again
so
A-1 A X = A-1 B
X =
│ 5/13 3/13││4│
│-1/13 2/13││2│
=
│2│
│0│
To solve the matrix equation AX = B for X, we can use the inverse matrix. The equation can be written as A⁻¹AX = A⁻¹B. Since A⁻¹A = I (identity matrix), this simplifies to X = A⁻¹B.
To find the inverse of matrix A, we first need to calculate the determinant of A. The determinant of a 2x2 matrix can be found by multiplying the elements on the main diagonal and subtracting the product of the elements on the opposite diagonal. For matrix A, the determinant is:
det(A) = (2*5) - (1*(-3))
= 10 + 3
= 13
Next, we calculate the inverse of A using the formula:
A⁻¹ = (1/det(A)) * adj(A), where adj(A) is the adjugate of A.
The adjugate of A is obtained by swapping the elements on the main diagonal and changing the sign of the elements on the opposite diagonal. For matrix A, the adjugate is:
adj(A) = [5 3]
[-1 2]
Now, we can find A⁻¹ by dividing adj(A) by the determinant:
A⁻¹ = (1/13) * [5 3]
[-1 2]
Multiplying A⁻¹ by B gives the solution for X:
X = A⁻¹B
= (1/13) * [5 3] * [4]
[2]
Performing the matrix multiplication:
X = (1/13) * [5*4 + 3*2]
[-1*4 + 2*2]
= (1/13) * [22]
[0]
Therefore, the solution to the equation AX = B where A = [2 -3; 1 5] and B = [4; 2] is X = [22/13; 0].