How to find the adjoint of a matrix?
Understand the Problem
The question is asking for the procedure to calculate the adjoint of a matrix, which involves finding the cofactor matrix and transposing it.
Answer
adjoint matrix
Answer for screen readers
The final answer is the adjoint matrix.
Steps to Solve
- Find the minor of each element in the matrix
To find the minor of an element in the matrix, remove the row and column containing that element and calculate the determinant of the resulting smaller matrix.
- Calculate the cofactor matrix
The cofactor of an element is the minor of the element multiplied by $(-1)^{i+j}$, where $i$ and $j$ are the row and column indexes of the element.
- Form the cofactor matrix
Create a matrix in which each element is replaced with its respective cofactor calculated in the previous step.
- Transpose the cofactor matrix
Switch the rows and columns of the cofactor matrix to obtain the adjoint matrix.
The final answer is the adjoint matrix.
More Information
The adjoint (or adjugate) is widely used in calculating the inverse of a matrix.
Tips
Common mistakes include miscalculating the minors and cofactors by not correctly using $(-1)^{i+j}$ or making errors in transposing the matrix.
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