An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Verify the matrix equation by performing matrix multiplication, addition, transposition, and inverse calculation.
Compare the left-hand side (LHS) and the right-hand side (RHS) of the equation.
After performing all the calculations, the equation holds true.
The given matrix equation is verified. T r u e
Explanation
Problem Analysis We are given three matrices A , B , and C , and we need to verify the matrix equation
3 A − 1 ( BC + B ) t − 2 \[ e wl in e ] ( A B − C t ) t = 3 \[ e wl in e ] ( A − 1 C t ) B t + 3 A − 1 B t − 2 B t A t + 2 C .
This involves calculating various matrix operations such as matrix multiplication, matrix addition, finding the inverse of a matrix, and transposing matrices. We will proceed step by step to verify the equation.
Verification Strategy The equation to verify is
3 A − 1 ( BC + B ) t − 2 ( A B − C t ) t = 3 ( A − 1 C t ) B t + 3 A − 1 B t − 2 B t A t + 2 C .
After performing all necessary calculations (matrix multiplication, addition, transposition, and inverse calculation), we compare both sides of the equation. If the left-hand side (LHS) equals the right-hand side (RHS), the equation is verified.
Result of Verification After performing all the matrix calculations, the result of comparing the left-hand side (LHS) and the right-hand side (RHS) of the given equation is:
The equation holds true.
Final Answer The given matrix equation
3 A − 1 ( BC + B ) t − 2 ( A B − C t ) t = 3 ( A − 1 C t ) B t + 3 A − 1 B t − 2 B t A t + 2 C
is verified to be true.
Examples
Matrix equations are fundamental in various fields such as computer graphics, physics, and engineering. For instance, in computer graphics, transformations like scaling, rotation, and translation of 3D objects are represented using matrices. Verifying matrix equations ensures that these transformations are applied correctly, preventing visual artifacts or errors in simulations. In structural engineering, matrix equations are used to analyze the stability and stress distribution in complex structures. Accurate verification of these equations is crucial to ensure the safety and reliability of the designed structures.
