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Linear Algebra Decoded
 

Linear Algebra Decoded Example

Linear Algebra Decoded
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This example has been taken directly from the solution given by Linear Algebra Decoded to the formulated problem.
Calculate the determinant of the matrix A defined as:

    ┌               ┐
    │  5   0   1  4 │
    │ -2   0  -3  2 │
A = │  2   0   0  2 │
    │ -2  -1  -1  2 │
    └               ┘

How to solve this problem?
The determinant can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products, where the cofactor of the entry located on i-th row and j-th column, is defined to be the determinant of the submatrix that remains after the i-th row and j-th column are deleted from the matrix, changing the sign if i+j is odd. From the previous definition it could be seen that cofactors involve determinants of lower order. Using this technique recursively together with the formulas for determinants of order 2 and 3, we have a method for calculating the determinant. In practice when expanding cofactors along a row or column to calculate the determinant, use the row or column with the greatest amount of zeros, because it is not needed to compute the associated cofactors. When a matrix is triangular, its determinant is the product of the entries on the main diagonal of the matrix.

Step 1: Expanding the cofactors along the row or column with the greater amount of zeros, because it reduces the volume of the calculations, since the cofactors associated to the zero entries should not be calculated. Determinants of order 3 are calculated using the formula: |A| = a11 a22 a33 + a12 a23 a 31 + a13 a21 a32 - a13 a22 a31 - a12 a21 a33 - a11 a23 a32.

│  5   0   1  4 │
│ -2   0  -3  2 │        │  5   1  4 │
│  2   0   0  2 │ =  - 1•│ -2  -3  2 │
│ -2  -1  -1  2 │        │  2   0  2 │

        │  5   1  4 │
        │ -2  -3  2 │ = 5•(-3)•2 + 1•2•2 + (-2)•0•4 - 4•(-3)•2 - 1•(-2)•2 - 2•0•5 = 2
        │  2   0  2 │

│  5   0   1  4 │
│ -2   0  -3  2 │
│  2   0   0  2 │ =  - 1•2 = -2
│ -2  -1  -1  2 │


Step 2: Final solution.

|A| = -2

  
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