3.9.3 A trick for calculating determinants a b • Consider the 2x2 matrix A = c d • Add the second column to the first and calculate the determinant: a+b b c+d d = (a+b)d –b(c+d) = (ad+bd) –(bc+bd) = (ad–bc) =|A| 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 • Example: 1 A= a 1 b 1 c C2 b+c a+c a+b • So, using the top row: | A | = (b-a)(a-c) – (c-a)(a-b) = ba-a2+ac-bc -(ac-bc -a2+ab) =0 C2-C1 1 a 0 1 b-a c b+c a-b a+b C3 1 a C3-C1 0 0 b-a c-a b+c a-b a-c 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: a11 0 0 a11 a12 a13 a11 0 0 0 a22 0 = 0 a22 a23 = a21 a22 0 = a11a22a33 0 0 a33 0 0 a33 a31 a32 a33 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 kb a b = k c d c d ka kb a b 2 = k kc kd c d 3.9.4 More determinant properties • If we swap two rows (or two columns), the determinant changes by a factor of (-1): c d a b a b = (-1) c d • If an entire row or column is zero, the determinant is zero 1 2 0 0 -1 -1 0 7 4 5 = 3 0 5 =0 0 0 1 0 4 3.9.4 More determinant properties • The determinant of a product is the product of determinants: |A B| = |A| |B| • Example 1 2 A= 3 4 -1 2 AB= -1 6 1 2 B= -1 0 |A| = -2, |B| = 2 So, |A B| = -4 = (-2)(2) = |A||B| 3.9.5 Cross product as determinant a1 • Consider two vectors: a = a2 a3 • Cross product is given by b1 b = b2 b3 i j k a x b = a 1 a2 a3 b 1 b2 b3 • Where i, j and k are unit vectors in the x,y and z directions. • Notice that a x b = - b x a