A matrix P (2 -1),Q(a) and R(2).Find the values
-4 3. b. 1.
of a and b given that PQ=R
To find the values of a and b, we can multiply matrices P and Q and equate it to matrix R.
We have matrix P as:
P = (2 -1)
(-4 3)
Matrix Q has one unknown value a:
Q = (a)
(b)
When we multiply matrices P and Q, we get:
PQ = (2a - b -a + 3b)
(-4a + 3b -4a + 9b)
We are given that PQ is equal to matrix R:
PQ = (2 -1)
(-4 3)
Equating corresponding elements, we get the following equations:
2a - b = 2 ----(1)
-a + 3b = -1 ----(2)
-4a + 3b = -4 ----(3)
-4a + 9b = 3 ----(4)
To solve these equations, we can use the method of substitution or elimination.
From equation (2), we get a = 3b - 1. Substituting this value in equation (1), we have:
2(3b - 1) - b = 2
6b - 2 - b = 2
5b - 2 = 2
5b = 4
b = 4/5
Substituting b = 4/5 in equation (2), we have:
-a + 3(4/5) = -1
-a + 12/5 = -1
-a = -1 - 12/5
-a = -5/5 - 12/5
-a = -17/5
a = 17/5
Therefore, the values of a and b are a = 17/5 and b = 4/5, respectively.
To find the values of a and b, we need to multiply matrix P with matrix Q and compare it with matrix R.
Matrix multiplication is done by multiplying the corresponding elements of the rows of the first matrix with the columns of the second matrix, and then summing up the products.
Given P = (2 -1)
-4 3
Q = (a)
(b)
R = (2)
To find PQ (the product of P and Q), we can multiply each element of P with the corresponding element of Q and sum them up.
PQ = (2 -1)(a) = (2a - a)
(-4 3) (b) (-4b + 3b)
Simplifying, we have:
PQ = (2a - a)
(-4b + 3b)
PQ = (a)
(-b)
Now, we can equate PQ with R:
(a) = (2)
(-b)
From here, we can determine the values of a and b:
a = 2
-b = 0
To find b, we multiply -1 to both sides of the equation -b = 0:
-b * -1 = 0 * -1
b = 0
Therefore, the values of a and b are:
a = 2
b = 0