Determine whether [tex]$B^{-1}$[/tex], the inverse of [tex]$B$[/tex], exists. Justify your answer. (Do not determine [tex]$B^{-1}$[/tex]).
[tex]$B=\left[\begin{array}{ccc}
1 & 2 & -1 \\
0 & 3 & 4 \\
1 & 7 & 2
\end{array}\right]$[/tex]
Answer (2)
Calculate the determinant of matrix B : d e t ( B ) = 1 ( 3 × 2 − 4 × 7 ) − 2 ( 0 × 2 − 4 × 1 ) + ( − 1 ) ( 0 × 7 − 3 × 1 ) .
Simplify the expression: d e t ( B ) = 1 ( 6 − 28 ) − 2 ( 0 − 4 ) − 1 ( 0 − 3 ) = − 22 + 8 + 3 = − 11 .
Since d e t ( B ) = − 11 = 0 , the inverse of B exists.
Therefore, B − 1 exists: T r u e .
Explanation
Problem Analysis and Strategy We are given the matrix B = 1 0 1 2 3 7 − 1 4 2 and we want to determine if the inverse of B , denoted as B − 1 , exists without actually calculating B − 1 . A matrix has an inverse if and only if its determinant is non-zero. Therefore, we need to calculate the determinant of B .
Calculating the Determinant The determinant of a 3x3 matrix a d g b e h c f i is given by a ( e i − f h ) − b ( d i − f g ) + c ( d h − e g ) .
Applying this to matrix B , we have: d e t ( B ) = 1 ∗ ( 3 ∗ 2 − 4 ∗ 7 ) − 2 ∗ ( 0 ∗ 2 − 4 ∗ 1 ) + ( − 1 ) ∗ ( 0 ∗ 7 − 3 ∗ 1 ) d e t ( B ) = 1 ∗ ( 6 − 28 ) − 2 ∗ ( 0 − 4 ) + ( − 1 ) ∗ ( 0 − 3 ) d e t ( B ) = 1 ∗ ( − 22 ) − 2 ∗ ( − 4 ) + ( − 1 ) ∗ ( − 3 ) d e t ( B ) = − 22 + 8 + 3 d e t ( B ) = − 11
Conclusion Since the determinant of B is − 11 , which is not equal to zero, the inverse of B , B − 1 , exists.
Examples
In computer graphics, matrices are used to represent transformations such as rotations, scaling, and translations of objects in 3D space. If a matrix representing a transformation has an inverse, it means that the transformation can be reversed. Determining whether a matrix has an inverse is crucial for ensuring that objects can be transformed back to their original positions or orientations. This is essential for interactive applications where users might want to undo transformations or manipulate objects in a reversible manner.
The inverse of the matrix B exists because its determinant, calculated to be -11, is not equal to zero. A matrix has an inverse if its determinant is non-zero, and in this case, it satisfies that condition. Thus, B − 1 exists.
;
