Determinants Artem Los (arteml@kth.se) February 6th, 2017 Artem - PowerPoint PPT Presentation

Determinants Artem Los (arteml@kth.se) February 6th, 2017 Artem Los (arteml@kth.se) Determinants February 6th, 2017 1 / 16

Overview What is a Determinant? 1 Rules and Theorems 2 Area of a Parallelogram 3 Artem Los (arteml@kth.se) Determinants February 6th, 2017 2 / 16

What is a Determinant? Artem Los (arteml@kth.se) Determinants February 6th, 2017 3 / 16

Definition Determinant In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix . (from https://en.wikipedia.org/wiki/Determinant ) Artem Los (arteml@kth.se) Determinants February 6th, 2017 4 / 16

Definition Determinant In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix . (from https://en.wikipedia.org/wiki/Determinant ) For example, Deducing if matrix A is invertible, i.e. ∃ A − 1 Finding area of a parallelogram Calculating eigenvalues (next week) Artem Los (arteml@kth.se) Determinants February 6th, 2017 4 / 16

Computing determinant The neat property of determinants is that we can always express a determinant of matrix n × n using determinants of size ( n − 1) × ( n − 1). Artem Los (arteml@kth.se) Determinants February 6th, 2017 5 / 16

Computing determinant The neat property of determinants is that we can always express a determinant of matrix n × n using determinants of size ( n − 1) × ( n − 1). Base case. 2 × 2 � � a b � � � = ad − bc � � c d � Artem Los (arteml@kth.se) Determinants February 6th, 2017 5 / 16

Computing determinant The neat property of determinants is that we can always express a determinant of matrix n × n using determinants of size ( n − 1) × ( n − 1). Base case. 2 × 2 � � a b � � � = ad − bc � � c d � Increasing dimension. 3 × 3 � � a b c � � � � � � � � e f d f d e � � � � � � � � = a � − b � + c d e f � � � � � � � � h i j i j h � � � � � � j h i � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 5 / 16

Computing determinant The neat property of determinants is that we can always express a determinant of matrix n × n using determinants of size ( n − 1) × ( n − 1). Base case. 2 × 2 � � a b � � � = ad − bc � � c d � Increasing dimension. 3 × 3 � � a b c � � � � � � � � e f d f d e � � � � � � � � = a � − b � + c d e f � � � � � � � � h i j i j h � � � � � � j h i � � � � a b c � � � � � � � � e f b c b c � � � � � � � � d e f = a � − d � + j � � � � � � � � h i h i e f � � � � � � j h i � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 5 / 16

Determine sign of the coefficients Question. In the example below, why is a positive but b negative? � � a b c � � � � � � � � e f d f d e � � � � � � � � = a � − b � + c d e f � � � � � � � � h i j i j h � � � � � � j h i � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 6 / 16

Determine sign of the coefficients Question. In the example below, why is a positive but b negative? � � a b c � � � � � � � � e f d f d e � � � � � � � � = a � − b � + c d e f � � � � � � � � h i j i j h � � � � � � j h i � � For 3 × 3, we have that:  + − +  − + −   + − + Artem Los (arteml@kth.se) Determinants February 6th, 2017 6 / 16

Determine sign of the coefficients Question. In the example below, why is a positive but b negative? � � a b c � � � � � � � � e f d f d e � � � � � � � � = a � − b � + c d e f � � � � � � � � h i j i j h � � � � � � j h i � � For 3 × 3, we have that:  + − +  − + −   + − + In general, the coefficient is ( − 1) i + j for element in i th row and j th column Artem Los (arteml@kth.se) Determinants February 6th, 2017 6 / 16

Example Problem. Compute the determinant of A, defined as:   1 2 1 A = 1 3 3   0 0 6 Artem Los (arteml@kth.se) Determinants February 6th, 2017 7 / 16

Example Problem. Compute the determinant of A, defined as:   1 2 1 A = 1 3 3   0 0 6 Step 1: ”Golden rule of determinant calculations: be lazy”. Pick the row/column with most zeros. Why? � � 1 2 1 � � � � 1 2 � � � � A = 1 3 3 = 6 � = 6(3 − 2) = 6 � � � � 1 3 � � � 0 0 6 � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 7 / 16

Rules and Theorems Artem Los (arteml@kth.se) Determinants February 6th, 2017 8 / 16

Rules (idea) Idea. Determinant matrix reductions are carried out to make it easier to figure out the determinant. The rules differ from elementary row operations. Artem Los (arteml@kth.se) Determinants February 6th, 2017 9 / 16

Rules (idea) Idea. Determinant matrix reductions are carried out to make it easier to figure out the determinant. The rules differ from elementary row operations. Goal. Get as many zeros as possible, for then we get less terms to compute. Artem Los (arteml@kth.se) Determinants February 6th, 2017 9 / 16

Rules (continued) Constant term factorization. det B = c det A � � � � c ∗ 1 c ∗ 2 c ∗ 3 1 2 3 � � � � � � � � 4 5 6 = c 4 5 6 � � � � � � � � 7 8 9 7 8 9 � � � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 10 / 16

Rules (continued) Constant term factorization. det B = c det A � � � � c ∗ 1 c ∗ 2 c ∗ 3 1 2 3 � � � � � � � � 4 5 6 = c 4 5 6 � � � � � � � � 7 8 9 7 8 9 � � � � Swapping two rows. det B = − det A � � � � 1 2 3 4 5 6 � � � � � � � � 4 5 6 = − 1 2 3 � � � � � � � � 7 8 9 7 8 9 � � � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 10 / 16

Rules (continued) Constant term factorization. det B = c det A � � � � c ∗ 1 c ∗ 2 c ∗ 3 1 2 3 � � � � � � � � 4 5 6 = c 4 5 6 � � � � � � � � 7 8 9 7 8 9 � � � � Swapping two rows. det B = − det A � � � � 1 2 3 4 5 6 � � � � � � � � 4 5 6 = − 1 2 3 � � � � � � � � 7 8 9 7 8 9 � � � � Adding two rows/columns. det B = det A � � � � 1 1 3 1 1 3 � � � � � � � � 1 1 6 = { R 2 : R 2 − R 1 } = 0 0 6 � � � � � � � � 7 8 9 7 8 9 � � � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 10 / 16

Theorems Theorem. Let A be a square matrix, i.e. n × n . Then: det A = det A T Artem Los (arteml@kth.se) Determinants February 6th, 2017 11 / 16

Theorems Theorem. Let A be a square matrix, i.e. n × n . Then: det A = det A T det A − 1 = 1 det A Artem Los (arteml@kth.se) Determinants February 6th, 2017 11 / 16

Theorems Theorem. Let A be a square matrix, i.e. n × n . Then: det A = det A T det A − 1 = 1 det A det AB = det A det B Artem Los (arteml@kth.se) Determinants February 6th, 2017 11 / 16

Theorems Theorem. Let A be a square matrix, i.e. n × n . Then: det A = det A T det A − 1 = 1 det A det AB = det A det B If two rows are equal = ⇒ det A = 0 Artem Los (arteml@kth.se) Determinants February 6th, 2017 11 / 16

Theorems Theorem. Let A be a square matrix, i.e. n × n . Then: det A = det A T det A − 1 = 1 det A det AB = det A det B If two rows are equal = ⇒ det A = 0 Theorem. Given that A is a square matrix, i.e. n × n , the following statements are equivalent: det A � = 0 rank( A ) = n A is invertible Artem Los (arteml@kth.se) Determinants February 6th, 2017 11 / 16

Example Problem. Find det A T A given that A is defined as shown below:   1 1 2 0 1 0 3 1   A =   2 3 0 0   0 0 1 4 Artem Los (arteml@kth.se) Determinants February 6th, 2017 12 / 16

Example Problem. Find det A T A given that A is defined as shown below:   1 1 2 0 1 0 3 1   A =   2 3 0 0   0 0 1 4 Step 1: (using known theorem): det ( A T A ) = (det A ) 2 . Artem Los (arteml@kth.se) Determinants February 6th, 2017 12 / 16

Example Problem. Find det A T A given that A is defined as shown below:   1 1 2 0 1 0 3 1   A =   2 3 0 0   0 0 1 4 Step 1: (using known theorem): det ( A T A ) = (det A ) 2 . Step 2: Find det A . � � 1 1 2 0 � � � � 1 0 3 1 � � = � � 2 3 0 0 � � � � 0 0 1 4 � � Artem Los (arteml@kth.se) Determinants February 6th, 2017 12 / 16