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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