Find the inverse of the matrix below.
[tex]\begin{array}{c}{\left[\begin{array}{cc}14 & 5 \\6 & 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 = ( 14 ) ( 3 ) − ( 5 ) ( 6 ) = 12 .
Find the inverse matrix using the formula: A − 1 = d e t 1 [ d − c − b a ] .
Substitute the values: A − 1 = 12 1 [ 3 − 6 − 5 14 ] = [ 4 1 2 − 1 12 − 5 6 7 ] .
Round the elements to the nearest hundredth: [ 0.25 − 0.5 − 0.42 1.17 ] .
Explanation
Problem Analysis We are given a 2x2 matrix and asked to find its inverse. The matrix is [ 14 6 5 3 ] We will follow the standard procedure for finding the inverse of a 2x2 matrix.
Calculating the Determinant First, we need to calculate the determinant of the matrix. For a matrix [ a c b d ] the determinant is given by a d − b c . In our case, a = 14 , b = 5 , c = 6 , and d = 3 . Therefore, the determinant is d e t = ( 14 ) ( 3 ) − ( 5 ) ( 6 ) = 42 − 30 = 12 Since the determinant is non-zero, the inverse exists.
Finding the Inverse Matrix Now, we find the inverse of the matrix. The inverse of a 2x2 matrix is given by d e t 1 [ d − c − b a ] In our case, the determinant is 12, so the inverse is 12 1 [ 3 − 6 − 5 14 ] = [ 12 3 12 − 6 12 − 5 12 14 ] = [ 4 1 2 − 1 12 − 5 6 7 ]
Rounding to Nearest Hundredth Finally, we convert the fractions to decimals and round to the nearest hundredth: [ 0.25 − 0.5 − 0.41666... 1.16666... ] ≈ [ 0.25 − 0.5 − 0.42 1.17 ] Thus, the inverse of the given matrix, rounded to the nearest hundredth, is [ 0.25 − 0.5 − 0.42 1.17 ]
Final Answer The inverse of the matrix is: [ 0.25 − 0.5 − 0.42 1.17 ]
Examples
In computer graphics, matrices are used to perform transformations such as rotations, scaling, and translations of objects. To undo a transformation, you need to find the inverse of the transformation matrix. For example, if you rotate an object by a certain angle using a matrix, you can use the inverse of that matrix to rotate the object back to its original position. This is crucial in creating interactive and dynamic visual experiences.
The inverse of the matrix A = [ 14 6 5 3 ] is A − 1 = [ 0.25 − 0.5 − 0.42 1.17 ] . This was calculated by first determining the determinant and then applying the formula for the inverse of a 2x2 matrix. The results were rounded to the nearest hundredth.
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