To find the inverse of a \(4 \times 4\) matrix \(A\), you can use several methods, such as the adjugate method, row reduction, or applying the formula with minors and cofactors. Below, I'll illustrate the method of row reduction to obtain the inverse of a matrix. Here’s how to do it step-by-step:
Step-by-Step Process to Find the Inverse of a 4x4 Matrix
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Write the Matrix Along with the Identity Matrix: Start with the matrix \(A\) and the \(4 \times 4\) identity matrix \(I\).
\[ \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & | & 1 & 0 & 0 & 0 \ a_{21} & a_{22} & a_{23} & a_{24} & | & 0 & 1 & 0 & 0 \ a_{31} & a_{32} & a_{33} & a_{34} & | & 0 & 0 & 1 & 0 \ a_{41} & a_{42} & a_{43} & a_{44} & | & 0 & 0 & 0 & 1 \ \end{bmatrix} \]
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Row Reduce the Augmented Matrix: Use elementary row operations to transform the left side into the identity matrix. The right side will become the inverse of \(A\) if \(A\) is invertible.
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Achieving Row Echelon Form:
- Swap rows if necessary to get a non-zero leading coefficient in the first row.
- Scale rows to make the first entry (leading coefficient) of each row to be 1.
- Use the leading 1's to clear out the entries below it (make them 0).
- Move to the next row and repeat the process until you transform the left side into the identity matrix.
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Reading off the Inverse: Once the left side is the identity matrix, the right side will become \(A^{-1}\).
Example
Let's find the inverse of the matrix:
\[ A = \begin{bmatrix} 4 & 3 & 2 & 1 \ 3 & 2 & 1 & 0 \ 2 & 1 & 0 & 1 \ 1 & 0 & 1 & 2 \end{bmatrix} \]
You can also use computer software or a calculator capable of handling matrix operations for larger matrices, but if you would like, I can guide you through the calculation step by step here, or I can help you write out the process via code in Python or another language. Would that work for you?