Find the inverse of the matrix below.
[tex]\begin{array}{c}{\left[\begin{array}{cc}16 & 5 \\8 & 3\end{array}\right]} \\{\left[\begin{array}{ll}0.38 & {[?]} \\
\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 ) = ( 16 ) ( 3 ) − ( 5 ) ( 8 ) = 8 .
Calculate the inverse of the matrix: A − 1 = 8 1 [ 3 − 8 − 5 16 ] = [ 0.375 − 1 − 0.625 2 ] .
Round the entries to the nearest hundredth: A − 1 = [ 0.38 − 1 − 0.63 2 ] .
The inverse matrix is: [ 0.38 − 1 − 0.63 2 ] .
Explanation
Problem Analysis We are given a 2x2 matrix and asked to find its inverse, rounding to the nearest hundredth if necessary. The matrix is A = [ 16 8 5 3 ] We are given a partially filled inverse matrix: A − 1 = [ 0.38 □ ? □ ] Our goal is to find the missing entries.
Calculating the Inverse The inverse of a 2x2 matrix A = [ a c b d ] is given by A − 1 = a d − b c 1 [ d − c − b a ] First, we need to calculate the determinant of A: d e t ( A ) = ( 16 ) ( 3 ) − ( 5 ) ( 8 ) = 48 − 40 = 8 Then, we can find the inverse of A: A − 1 = 8 1 [ 3 − 8 − 5 16 ] = [ 3/8 − 8/8 − 5/8 16/8 ] = [ 0.375 − 1 − 0.625 2 ]
Finding Missing Entries Now, we round the entries of the inverse matrix to the nearest hundredth: A − 1 = [ 0.38 − 1 − 0.63 2 ] Finally, we fill in the missing entries in the given inverse matrix: A − 1 = [ 0.38 − 1 − 0.63 2 ]
Examples
In computer graphics, matrices and their inverses are used to perform transformations such as rotations, scaling, and translations of objects in 2D or 3D space. Finding the inverse of a transformation matrix allows you to undo a transformation, which is useful for tasks like object manipulation and camera control. For example, if you rotate an object by a certain angle, you can use the inverse of the rotation matrix to rotate it back to its original orientation.
The inverse of the matrix A = [ 16 8 5 3 ] is A − 1 = [ 0.38 − 1 − 0.63 2 ] , after calculating the determinant and appropriate adjustments. Each entry is rounded to the nearest hundredth as requested.
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