- Use the definition of the inverse matrix \(C \cdot C^{-1} = I\) to generate a system of equations.
- Solve the resulting system \(-b + 4c = 0\), \(6a - 2c = 0\), and \(a = 1\) through substitution.
- Determine the values as \(a=1\), \(b=12\), and \(c=3\).
- Final results: \(a = \boxed{1}\), \(b = \boxed{12}\), \(c = \boxed{3}\).
Explanation
- Problem Analysis and Setup
The goal is to determine the values of \(a\), \(b\), and \(c\) such that the given matrix \(C = \begin{bmatrix} 0 & -1 & 4 \ 6 & 0 & -2 \ 1 & 0 & 0 \end{bmatrix}\) satisfies \(C \cdot C^{-1} = I\), where \(C^{-1} = \begin{bmatrix} 0 & 0 & a \ -1 & -2 & b \ 0 & -\frac{1}{2} & c \end{bmatrix}\) and \(I\) is the \(3 \times 3\) identity matrix. We perform matrix multiplication and equate the resulting product to the identity matrix.
- Matrix Multiplication and Solving for Variables
We multiply \(C\) and \(C^{-1}\):
\[\begin{bmatrix} 0 & -1 & 4 \ 6 & 0 & -2 \ 1 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 & a \ -1 & -2 & b \ 0 & -\frac{1}{2} & c \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}.\]
Computing the product row by row:
First row, third column: \(0 \cdot a + (-1) \cdot b + 4 \cdot c = 0 \Rightarrow -b + 4c = 0\).
Second row, third column: \(6 \cdot a + 0 \cdot b + (-2) \cdot c = 0 \Rightarrow 6a - 2c = 0\).
Third row, third column: \(1 \cdot a + 0 \cdot b + 0 \cdot c = 1 \Rightarrow a = 1\).
Also, checking the entries for the first and second columns verifies the structure:
Third row, first column: \(1 \cdot 0 + 0 \cdot (-1) + 0 \cdot 0 = 0\).
Third row, second column: \(1 \cdot 0 + 0 \cdot (-2) + 0 \cdot (-\frac{1}{2}) = 0\).
This confirms the validity of the structure for \(C^{-1}\).
- Final Values calculation
From the equation \(a = 1\), we substitute this into \(6a - 2c = 0\):
\[6(1) - 2c = 0 \Rightarrow 6 - 2c = 0 \Rightarrow 2c = 6 \Rightarrow c = 3.\]
Now substitute \(c = 3\) into the equation \(-b + 4c = 0\):
\[-b + 4(3) = 0 \Rightarrow -b + 12 = 0 \Rightarrow b = 12.\]
Thus, the values are \(a = 1\), \(b = 12\), and \(c = 3\).
Examples
Matrix inversion is fundamental in engineering and computer science, particularly in solving systems of linear equations. For example, when calculating the structural response of a building under wind loads or in computer graphics when transforming 3D objects, representing these transformations as matrices and finding their inverses allows us to reverse the operations to return to an original state or solve for unknown forces and displacements. Understanding matrix inversion provides the foundation for these complex analytical processes.