Find the matrix B such that AB=t and A=(3 2)
(-1 3)
Hence find the point of intersection of the lines 3x+2y=10 and 3y-4x=x
To find matrix B, we need to solve the equation AB = t, where A is given as:
A = |3 2|
|-1 3|
And t is the desired product.
We are given that t = |3x+2y|
|3y-4x = x|
We need to multiply A by B such that the product satisfies the given equations. Let's expand the matrix multiplication:
|3 2| |x| |3x+2y|
| | * | | = t
|-1 3| |y| |3y-4x|
So, we need to find the values of x and y that satisfy the above equation. Comparing the elements of the two matrices, we get the following system of equations:
3x + 2y = 3x + 2y ... (1)
3y - 4x = 3y - 4x + x ... (2)
Equation (1) is always true, so it does not provide any extra information. Equation (2) simplifies to:
-4x = x => x = 0
Substituting the value of x into equation (2):
3y - 4(0) = 3y = 0
y = 0
Therefore, the point of intersection of the lines 3x + 2y = 10 and 3y - 4x = x is (0, 0).
Now, let's find matrix B. Since x = 0 and y = 0, we have:
|3 2| |0| |0|
| | * | | = |0|
|-1 3| |0| |0|
So, matrix B is:
B = |0|
|0|
To find the matrix B such that AB = t, we need to solve the equation using matrix algebra.
Given A = (3 2)
(-1 3)
Let's assume B = (a b)
(c d)
Then, AB will be equal to:
AB = (3 2) (a b) = (3a + 2c, 3b + 2d)
(-1 3) (c d) (-a + 3c, -b + 3d)
We are given that AB = t, but we don't know the value of t. So, we will look at the components of t to equate them with the corresponding components in AB.
Let's assume t = (x y)
Equating components, we get the following equations:
3a + 2c = x
3b + 2d = y
-a + 3c = x
-b + 3d = y
Now, we can solve these equations to find the values of a, b, c, and d.
Adding the equations 3a + 2c = x and -a + 3c = x, we get:
2a + 5c = 2x (equation 1)
Adding the equations 3b + 2d = y and -b + 3d = y, we get:
2b + 5d = 2y (equation 2)
From equation 1, we can express a in terms of c:
a = (2x - 5c)/2
Substituting this into equation 2, we can express b in terms of d:
b = (2y - 5d)/2
Therefore, the matrix B can be expressed as:
B = (a b) = ((2x - 5c)/2, (2y - 5d)/2)
(c d) (c, d)
Now, let's find the point of intersection of the lines 3x + 2y = 10 and 3y - 4x = x.
To find the point of intersection, we can solve the system of equations formed by these two equations:
3x + 2y = 10 (equation 3)
3y - 4x = x (equation 4)
Simplifying equation 4, we get:
3y - 4x = x
3y = 5x
y = (5/3)x
Substituting this value of y into equation 3, we get:
3x + 2(5/3)x = 10
3x + 10x/3 = 10
9x + 10x = 30
19x = 30
x = 30/19
Substituting this value of x back into equation 4, we get:
3y - 4(30/19) = 30/19
3y = (120/19) + (30/19)
3y = 150/19
y = (50/19)
Therefore, the point of intersection of the lines is (30/19, 50/19).