Given that [tex]P =\left(\begin{array}{ll} 5 & 3 \\ 6 & 4 \end{array}\right)[/tex], find:
a. Its inverse (1 mk)
b. The value of x and y if [tex]P \binom{x}{y}=\binom{3}{2}[/tex] (3mks)
Answer (2)
The inverse of the matrix P is P − 1 = ( 2 − 3 − 2 3 2 5 ) , and the values of x and y are x = 3 and y = − 4 .
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Calculate the determinant of matrix P: d e t ( P ) = ( 5 ) ( 4 ) − ( 3 ) ( 6 ) = 2 .
Find the inverse of P: P − 1 = 2 1 ( 4 − 3 − 6 5 ) = ( 2 − 2 3 − 3 2 5 ) .
Multiply P − 1 by ( 2 3 ) : ( y x ) = ( 2 − 2 3 − 3 2 5 ) ( 2 3 ) = ( − 4 3 ) .
The solution is x = 3 and y = − 4 , and the inverse matrix is P − 1 = ( 2 − 2 3 − 3 2 5 ) . x = 3 , y = − 4 , P − 1 = ( 2 − 2 3 − 3 2 5 )
Explanation
Problem Analysis We are given the matrix P = ( 5 3 6 4 ) and asked to find its inverse and solve the equation P ( y x ) = ( 2 3 ) for x and y .
Finding the Determinant First, we need to find the inverse of matrix P . The determinant of P is given by d e t ( P ) = ( 5 ) ( 4 ) − ( 3 ) ( 6 ) = 20 − 18 = 2 . Since the determinant is non-zero, the inverse exists.
Calculating the Inverse The inverse of a 2x2 matrix ( a b c d ) is given by a d − b c 1 ( d − b − c a ) . Therefore, the inverse of P is: P − 1 = 2 1 ( 4 − 3 − 6 5 ) = ( 2 − 2 3 − 3 2 5 )
Setting up the Equation Now, we need to solve for x and y in the equation P ( y x ) = ( 2 3 ) . We can do this by multiplying both sides of the equation by P − 1 :
( y x ) = P − 1 ( 2 3 ) = ( 2 − 2 3 − 3 2 5 ) ( 2 3 )
Solving for x and y Performing the matrix multiplication, we get: ( y x ) = ( ( 2 ) ( 3 ) + ( − 2 3 ) ( 2 ) ( − 3 ) ( 3 ) + ( 2 5 ) ( 2 ) ) = ( 6 − 3 − 9 + 5 ) = ( 3 − 4 ) Thus, x = 3 and y = − 4 .
Final Answer Therefore, the inverse of P is P − 1 = ( 2 − 2 3 − 3 2 5 ) and the values of x and y are x = 3 and y = − 4 .
Examples
Matrix operations, like finding the inverse and solving systems of equations, are crucial in computer graphics for transformations such as rotating, scaling, and translating objects in 3D space. For instance, if you want to rotate an object in a game, you can represent the object's vertices as a matrix and multiply it by a rotation matrix (similar to matrix P). Solving a system of equations can help determine how to combine different transformations to achieve a desired effect, ensuring that objects appear correctly on the screen. These concepts are also used in physics simulations to calculate forces and motions.
