1 / 8

Consider the 2x2 matrix

3.9.3 A trick for calculating determinants. A =. Consider the 2x2 matrix. Add the second column to the first and calculate the determinant:. = (a+b)d –b(c+d). a. b. a+b. b. = (ad+bd) –(bc+bd). c. d. c+d. d. = (ad– bc ) =| A |. 3.9.3 A trick for calculating determinants.

hyatt-rice
Download Presentation

Consider the 2x2 matrix

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3.9.3 A trick for calculating determinants A = • Consider the 2x2 matrix • Add the second column to the first and calculate the determinant: = (a+b)d –b(c+d) a b a+b b = (ad+bd) –(bc+bd) c d c+d d = (ad–bc) =|A|

  2. 3.9.3 A trick for calculating determinants • In fact if you replace any column of a matrix by the original column + a multiple of any other column the determinant is unchanged. • Similarly, if you replace any row of a matrix by the original row + a multiple of any other row the determinant is unchanged. • WARNING: adding one row or column to itself will in general change the determinant

  3. Example: 1 0 1 1 1 1 C2 C2-C1 A = a b c a b-a c b+c a+c a+b b+c a-b a+b • So, using the top row: C3 C3-C1 | A | = (b-a)(a-c) – (c-a)(a-b) = ba-a2+ac-bc -(ac-bc -a2+ab) = 0 0 1 0 a b-a c-a b+c a-b a-c

  4. 3.9.4 More determinant properties • If we take the transpose of a matrix, its determinant is unchanged: |A| = |AT| • For diagonal or upper triangular or lower triangular matrices, the determinant is the product of the leading diagonal entries: a13 0 0 a11 a11 a11 0 0 a12 = a11a22a33 a21 0 0 0 a23 0 = = a22 a22 a22 a32 0 0 0 0 a31 a33 a33 a33

  5. 3.9.4 More determinant properties • Multiplying a whole row (or column) by k multiplies the determinant by k. • If a matrix is nxn then multiplying the matrix by k is the same as multiplying n rows by k. Hence, the determinant is multiplied by kn. ka a a ka b kb kb b kc c c c d d kd d = k = k2

  6. 3.9.4 More determinant properties • If we swap two rows (or two columns), the determinant changes by a factor of (-1): • If an entire row or column is zero, the determinant is zero -1 7 c a d b 1 -1 0 0 3 2 5 5 a c b d 0 4 = (-1) = = 0 0 0 1 0 4 0

  7. 3.9.4 More determinant properties • The determinant of a product is the product of determinants: |AB| = |A| |B| • Example 1 2 1 2 |A| = -2, |B| = 2 A = B = 3 4 -1 0 -1 2 So, |AB| = -4 = (-2)(2) = |A||B| AB = -1 6

  8. 3.9.5 Cross product as determinant a1 b1 a = b = • Consider two vectors: a2 b2 a3 b3 • Cross product is given by a x b = a3 k i j b1 b2 b3 a1 a2 • Where i, j and k are unit vectors in the x,y and z directions. • Notice that a x b = - b x a

More Related