Evaluate the determinant for the following matrix:

ANSWER

The determinant is 0

EXPLANATION

For an n×n matrix A, the determinant of A, det(A), can be obtained by expanding along the kth row:

where is the entry of A in the kth row, 1st column, is the entry of A in the kth row, 2nd column, etc., and is the kn cofactor of A, defined as .

is the kn minor, obtained by getting the determinant of the matrix which is the matrix A with row k and column n deleted.

Applying this here, we can expand along the 1st row. For convenience, let G be the matrix given by

where the first row has been bolded.

The determinant of G is therefore

Note that g₁₁ is the matrix element of G that is in the 1st row, 1st column,

g₁₂ is the matrix element of G that is in the 1st row, 2nd column, etc. Then we have

M₁₁ is the determinant of the matrix that is matrix G with row 1 and column 1 removed. The bold entires are the row and the column we delete.

The determinant of a 2×2 matrix is

so it follows that

Applying the same for M₁₂ and M₁₃, we have

and

so therefore

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