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: .
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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