__PROPERTIES OF
DETERMINANTS__

Here is the same list of properties that is contained the previous lecture.

(1.) A multiple of one row of "A" is added to
another row to produce a matrix, "B", then:_{}.

(2.) If two rows are interchanged to produce a matrix,
"B", then:_{}.

(3.) If one row is multiplied by "k" to produce a
matrix, "B", then: _{}.

(4.) If "A" and "B" are both n x n
matrices then: _{}.

(5.)_{}.

__Example # 1__: Find the
determinant by row reduction to echelon form.

_{ }

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We now have "A" in upper triangular form. Since we have 3 pivots, "A" is invertible. If we continue the reduction process we could obtain a diagonal matrix.

_{ }

_{}

_{}

Evidently, we only needed to go as far as echelon form to identify the values of the pivots, since the determinant is their product.

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__Example # 2__: Find det
B given det A.

_{}

_{}

_{}

_{}

__Example # 3__: Find det
B given det A.

_{}

_{}

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_{}

_{}

__Example # 4__: Show
that if 2 rows of a square matrix "A" are the same, then det A = 0.

Suppose rows "i" and "j" are identical.
Then if we exchange those rows, we get the same matrix and thus the same
determinant. However, a row exchange changes the sign of the determinant. This
requires that _{},
which can only be true if _{}.

__Example # 5__: Use
determinants to decide if the set of vectors is linearly independent.

_{ }

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_{}

_{}

_{ }

The vectors are Linearly Dependent.

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__Example # 6__: Show
that if "A" is invertible, then _{}.

_{}

_{}

_{}

__Example # 7__: Show
that if "A" is invertible, then _{}.

_{}

_{}

The matrix "L" is lower triangular. Its transpose
is upper triangular. The determinants of upper and lower non-singular matrices
are the products of their diagonal elements. Since the transpose does not
change the diagonal elements, then _{} and _{}.

_{}

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