If the system below were written as a matrix equation, by which matrix could you multiply both sides to obtain a solution?
4x + 6y = 24
5x + 8y = 40
If we pre-multiply both sides of the equation by the inverse of the square matrix,
m=
4 6
5 8
b=
24
40
m-inverse=
4 -3
-5/2 2
and
m-inverse * B =
-24
20
we get an identity matrix on the left, and the solution vector on the right, i.e. x=-24, y=20.
To write the given system as a matrix equation, we can represent the coefficients of the variables and the constant terms in a matrix form.
First, let's represent the variables and constant terms in a matrix:
| 4 6 | | x | = | 24 |
| 5 8 | | y | | 40 |
Now, we can rewrite this matrix equation as:
A * X = B,
where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
By comparing the coefficients, we can see that the coefficient matrix A is:
A = | 4 6 |
| 5 8 |
And the constant matrix B is:
B = | 24 |
| 40 |
To obtain a solution, we need to multiply both sides of the equation by the inverse of matrix A. Multiplying by the inverse of A "undoes" the effect of the coefficient matrix and gives us the solution.
So, to obtain a solution, we need to multiply both sides of the equation A * X = B by the inverse of matrix A.