In this section, we associated a numerical quantity, the determinant, to a square matrix and showed how it tells us whether the matrix is invertible.
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The determinant of a matrix has a geometric interpretation. In particular, when
the determinant is the signed area of the parallelogram formed by the two columns of the matrix.
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The determinant satisfies many properties. For instance,
and the determinant of a triangular matrix is equal to the product of its diagonal entries.
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These properties helped us compute the determinant of a matrix using row operations. This also led to the important observation that the determinant of a matrix is nonzero if and only if the matrix is invertible.
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Finally, we learned how to compute the determinant of a matrix using cofactor expansions, which will be a valuable tool for us in the next chapter.
The geometric definition of the determinant tells us that the determinant is measuring a natural geometric quantity, an insight that does not easily come through the other two approaches. The intuition we gain by thinking about the determinant geometrically makes it seem reasonable that the determinant should be zero for matrices that are not invertible: if the columns are linearly dependent, the vectors cannot create a positive volume.
Approaching the determinant through row operations provides an effective means of computing the determinant. In fact, this is what most computer programs do behind the scenes when they compute a determinant. This approach is also a useful theoretical tool for explaining why the determinant tells us whether a matrix is invertible.
The cofactor expansion method will be useful to us in the next chapter when we look at eigenvalues and eigenvectors. It is not, however, a practical way to compute a determinant. To see why, consider the fact that the determinant of a
matrix, written as
requires us to compute two terms,
and
To compute the determinant of a
matrix, we need to compute three
determinants, which involves
terms. For a
matrix, we need to compute four
determinants, which produces
terms. Continuing in this way, we see that the cofactor expansion of a
matrix would involve
terms.
By contrast, we have seen that the number of steps required to perform Gaussian elimination on an
matrix is proportional to
When
we have
which points to the fact that finding the determinant using Gaussian elimination is considerably less work.