Find the inverse of the matrix below:
[tex]$\begin{array}{c}{\left[\begin{array}{cc}20 & 2 \\-40 & -3\end{array}\right]} \\{\left[\begin{array}{ll}{[?]} & \square \\ \square & \square\end{array}\right]}\end{array}$[/tex]
If necessary, round to the nearest hundredth.
Answer (2)
Calculate the determinant of the matrix: d e t ( A ) = ( 20 ) ( − 3 ) − ( 2 ) ( − 40 ) = 20 .
Compute the inverse using the formula: A − 1 = d e t ( A ) 1 [ − 3 40 − 2 20 ] .
Divide each entry of the matrix by the determinant: A − 1 = [ − 20 3 20 40 − 20 2 20 20 ] .
Simplify to find the inverse: A − 1 = [ − 0.15 2 − 0.1 1 ] .
Explanation
Problem Setup We are given the matrix A = [ 20 − 40 2 − 3 ] and we want to find its inverse.
Calculate the Determinant First, we need to calculate the determinant of the matrix A . The determinant is given by d e t ( A ) = ( 20 ) ( − 3 ) − ( 2 ) ( − 40 ) = − 60 + 80 = 20 .
Compute the Inverse Matrix Since the determinant is non-zero, the inverse exists. The inverse of a 2x2 matrix [ a c b d ] is given by a d − b c 1 [ d − c − b a ] . Therefore, the inverse of A is given by
A − 1 = 20 1 [ − 3 40 − 2 20 ] = [ − 20 3 20 40 − 20 2 20 20 ] = [ − 0.15 2 − 0.1 1 ] .
Final Answer Thus, the inverse of the given matrix is [ − 0.15 2 − 0.1 1 ] .
Examples
In computer graphics, matrix inverses are used to transform objects back to their original positions after a series of transformations (like rotations, scaling, and translations) have been applied. For example, if you rotate an object and then want to undo the rotation, you would use the inverse of the rotation matrix. This ensures that the object returns to its initial orientation, which is crucial for accurate rendering and interactive manipulation of 3D models.
The inverse of the matrix A = [ 20 − 40 2 − 3 ] is A − 1 = [ − 0.15 2 − 0.1 1 ] . This was calculated by first determining the determinant and then applying the inverse formula. Finally, we simplified the resulting matrix entries.
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