Which value of [tex]$x$[/tex] makes the two matrices inverses of each other?
[tex]$\left[\begin{array}{ll}
5 & 2 \\
2 & 1
\end{array}\right]\left[\begin{array}{cc}
1 & -2 \\
x & 5
\end{array}\right]$[/tex]
A. -2
B. -1
C. 1
D. 2
Answer (2)
To find the value of x that makes the two matrices inverses of each other, we multiply the matrices and set the result equal to the identity matrix. Solving the equations derived from this multiplication leads to the solution x = − 2 . Thus, the correct answer is -2.
;
Multiply the two matrices: [ 5 2 2 1 ] [ 1 x − 2 5 ] = [ 5 + 2 x 2 + x 0 1 ] .
Set the resulting matrix equal to the identity matrix: [ 5 + 2 x 2 + x 0 1 ] = [ 1 0 0 1 ] .
Solve the equation 2 + x = 0 for x , which gives x = − 2 .
The value of x that makes the two matrices inverses of each other is − 2 .
Explanation
Understanding the Problem We are given two matrices and asked to find the value of x that makes them inverses of each other. Two matrices are inverses if their product is the identity matrix. The identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. For a 2x2 matrix, the identity matrix is [ 1 0 0 1 ] .
Multiplying the Matrices To find the value of x , we need to multiply the two matrices and set the result equal to the identity matrix. Let's multiply the given matrices:
[ 5 2 2 1 ] [ 1 x − 2 5 ] = [ ( 5 ) ( 1 ) + ( 2 ) ( x ) ( 2 ) ( 1 ) + ( 1 ) ( x ) ( 5 ) ( − 2 ) + ( 2 ) ( 5 ) ( 2 ) ( − 2 ) + ( 1 ) ( 5 ) ] = [ 5 + 2 x 2 + x − 10 + 10 − 4 + 5 ] = [ 5 + 2 x 2 + x 0 1 ]
Setting Equal to the Identity Matrix Now, we set the resulting matrix equal to the identity matrix:
[ 5 + 2 x 2 + x 0 1 ] = [ 1 0 0 1 ]
Solving for x From the equation above, we can set up the following equations:
5 + 2 x = 1
2 + x = 0
Solving the second equation for x :
2 + x = 0 ⇒ x = − 2
Verifying the Solution Let's check if x = − 2 satisfies the first equation:
5 + 2 ( − 2 ) = 5 − 4 = 1 . So, x = − 2 satisfies both equations.
Final Answer Therefore, the value of x that makes the two matrices inverses of each other is − 2 .
Examples
Matrix inverses are used in computer graphics to transform objects back to their original positions after a series of transformations, such as rotations or scaling. They are also used in solving systems of linear equations, which can model various real-world problems, such as network flows or circuit analysis. Understanding matrix inverses helps in efficiently reversing processes and solving complex problems in engineering and computer science.
