A matrix P (2 -1 ),Q(a) and R(2).Find the
-4 3. b. 1.
values of a and b given that PQ=R
To find the values of a and b, we need to solve the equation PQ = R.
The product of two matrices is obtained by multiplying the corresponding elements and summing the products. In this case, we have:
PQ = (2 -1) * (a b) = (2a - b , -2a + 3b)
Since PQ = R, we have:
2a - b = 2
-2a + 3b = 1
We can solve this system of equations using either substitution or elimination method. Let's use the elimination method:
Multiply the first equation by 3 and the second equation by 2 to make the coefficients of b in both equations cancel out:
6a - 3b = 6
-4a + 6b = 2
Now, add the two equations:
6a - 3b + (-4a + 6b) = 6 + 2
2a + 3b = 8
Now, solve for a:
2a = 8 - 3b
a = (8 - 3b)/2
Substitute this expression for a in the first equation:
2(8 - 3b)/2 - b = 2
8 - 3b - b = 2
-4b = -6
b = -6/-4
b = 3/2 or 1.5
Now, substitute the value of b back into the expression for a:
a = (8 - 3(3/2))/2
a = (8 - 9/2)/2
a = (16/2 - 9/2)/2
a = 7/2 /2
a = 7/4 or 1.75
So, the values of a and b are a = 7/4 or 1.75 and b = 3/2 or 1.5.
To find the values of a and b given that PQ = R, we need to multiply matrix P by matrix Q and compare the resulting matrix with matrix R.
Let's start by multiplying matrix P by matrix Q:
P = [2 -1]
[-4 3]
Q = [a]
[b]
To multiply these matrices, we can use the row-by-column method.
The first element of the resulting matrix will be the dot product of the first row of P and the first column of Q. So the first element is:
(2 * a) + (-1 * b) = 2a - b
The second element of the resulting matrix will be the dot product of the second row of P and the first column of Q. So the second element is:
(-4 * a) + (3 * b) = -4a + 3b
Now, let's compare the resulting matrix with matrix R:
R = [2]
[1]
We have PQ = R, which means that the resulting matrix should be equal to R. So we can set up the following equations:
2a - b = 2
-4a + 3b = 1
We can solve this system of equations to find the values of a and b.
Using the second equation, we can solve for a:
-4a + 3b = 1
-4a = -3b + 1
a = (3b - 1) / -4
Now, we substitute this value of a into the first equation and solve for b:
2a - b = 2
2((3b - 1) / -4) - b = 2
(6b - 2) / -4 - b = 2
(6b - 2) - 4b = 8
6b - 2 - 4b = 8
2b - 2 = 8
2b = 10
b = 5
So, we have found that b = 5.
Now, substitute the value of b into the equation for a:
a = (3b - 1) / -4
a = (3 * 5 - 1) / -4
a = (15 - 1) / -4
a = 14 / -4
a = -7/2
Therefore, the values of a and b that satisfy PQ = R are a = -7/2 and b = 5.
To find the values of a and b given that PQ = R, we need to calculate the product of matrices P and Q and equate it to matrix R. Let's begin:
The product of matrices P (2 -1) and Q (a b) is calculated as follows:
PQ = (2 * a) + (-1 * 1) = 2a - 1
Now, we equate this product to matrix R (2 1):
2a - 1 = 2
To solve for a, we add 1 to both sides of the equation:
2a = 3
Dividing both sides by 2, we find:
a = 3/2 = 1.5
Now, let's calculate the product of matrices P (-4 3) and Q (a b):
PQ = (-4 * a) + (3 * 1) = -4a + 3
We equate this product to matrix R (2 1):
-4a + 3 = 1
To solve for b, we subtract 3 from both sides of the equation:
-4a = -2
Dividing both sides by -4, we find:
a = 1/2 = 0.5
Therefore, the values of a and b that satisfy PQ = R are a = 1.5 and b = -2.