DETERMINANTS, MINORS AND COFACTORS ALGEBRA 6 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Determinants, minors and cofactors 1/ 15 Adrian Jannetta
Determinants Every square matrix has a number called a determinant associated with it. The determinant is calculated using the elements of the matrix. Determinants have many applications in mathematics. They are useful for deciding whether a set of equations have consistent solutions (for example, in deciding whether or not vector lines are skew or intersecting). As you will see, the determinant is also necessary for calculating the inverse of a matrix. In this presentation we ll only consider determinants of 2 2 and 3 3 matrices. Determinants, minors and cofactors 2/ 15 Adrian Jannetta
Second order determinants The 2 2 matrix a b A= c d has a determinant which is calculated as follows: A = a b c d = ad bc Example Find the determinant of the matrix B= 3 1 2 4 In this case the determinant is B = 3 1 2 4 =(3 4) (1 2)=12+2=14. Determinants, minors and cofactors 3/ 15 Adrian Jannetta
Third order determinants Determinants of of matrices of size 3 3 (or larger) are evaluated differently. To calculate them we must include the appropriate sign (±) for each element. The signs of elements alternate between plus and minus like this: a b c + + d e f elements associated with signs + g h i + + A plus sign means leave the element unchanged. A minus means flip the sign of the element. The 3 3 determinant is first expressed in terms of the 2 2 determinants. Determinants, minors and cofactors 4/ 15 Adrian Jannetta
Minors Each element of a 3 3 matrix is associated with a 2 2 determinant called a minor. It is found by taking elements from the row and column not containing that element. Consider the matrix a b c d e f g h i a b c d e f g h i a b c d e f g h i a b c d e f g h i a b c d e f g h i Element a is associated with the 2 2 minor e h Element b is associated with the 2 2 minor d g Element c is associated with the 2 2 minor d g Element d is associated with the 2 2 minor b h f i. f i. e h. c i. Determinants,...and sominors on. and cofactors 5/ 15 Adrian Jannetta
3 3 determinants can be expanded along any row or column. For simplicity, let s see how to expand along the first row. We note that the signs are+ + in the expansion of the first row. a b c d e f g h i = a e h f i b d g f i + c d g e h Now we can expand each of the 2 2 determinants in the usual way: a b c d e f = a(ei fh) b(di fg)+c(dh eg) g h i This is not a formula to memorise try to learn the method of expansion. Let s try this on a matrix with actual values. Determinants, minors and cofactors 6/ 15 Adrian Jannetta
Evaluating a 3 3 determinant Find the determinant associated with the matrix 2 5 1 A= 9 7 4 0 3 6 Let us expand along the first row: 2 5 1 deta = 9 7 4 0 3 6 = 2 7 4 3 6 5 9 4 0 6 + 1 9 7 0 3 deta = 2(42 12) 5(54 0)+1(27 0) = 2(30) 5(54)+1(27) = 60 270+27 = 183 Determinants, minors and cofactors 7/ 15 Adrian Jannetta
Evaluating a 3 3 determinant 3 2 1 Evaluate 2 1 4 8 0 0 If we expand along the first row: 3 2 1 2 1 4 = +3 8 0 0 1 4 0 0 = 3(0)+2( 32)+8 = 64+8= 56 ( 2) 2 4 8 0 + 1 2 1 8 0 However, it is quicker to use the 3rd row (with those zeros!) 3 2 1 2 1 4 = +8 2 1 8 0 0 1 4 + 0+0 = 8( 7) = 56 Determinants, minors and cofactors 8/ 15 Adrian Jannetta
Cofactor matrix Given the matrix A= a b c d e f g h i we saw previously that each element is associated with a 2 2 determinant. The values of those determinants are called cofactors. If we calculate the values of all 9 determinants, taking into account the behaviour of a b c + + d e f elements associated with signs + g h i + + we can make a matrix from the cofactors.remember: a plus means leave the element unchanged. A minus means flip the sign of the element. Calculating the cofactor matrix will be an important step in calculating inverse of a matrix in the next presentation. Determinants, minors and cofactors 9/ 15 Adrian Jannetta
Cofactor matrix 1 3 2 Given A= 1 4 1 find the matrix of cofactors C. 2 2 5 Let s work along the top row of A to calculate the elements of the new matrix C. Remember: + leaves the sign unchanged, while flips it. C 11 =+ 4 1 2 5 =+( 20 2)= 22 C 12 = 1 1 2 5 = (5 2)= 3 C 13 =+ 1 4 2 2 =+( 2 8)= 10 So far the cofactor matrix looks like this: C= 22 3 10 We ll continue with the second row. Determinants, minors and cofactors 10/ 15 Adrian Jannetta
C 21 = C 22 =+ C 23 = 3 2 2 5 1 2 2 5 1 3 2 2 = ( 15+4)=11 =+( 5+4)= 1 = (2 6)=4 Now the cofactor matrix looks like this: 22 3 10 C= 11 1 4 And now the final row... Determinants, minors and cofactors 11/ 15 Adrian Jannetta
C 31 =+ C 32 = C 33 =+ 3 2 4 1 1 2 1 1 1 3 1 4 =+(3+8)=11 = (1 2)=1 =+(4+3)=7 The completed matrix of cofactors is 22 3 10 C= 11 1 4 11 1 7 The cofactor matrix is a crucial step in finding the inverse of a matrix later in the next presentation. Determinants, minors and cofactors 12/ 15 Adrian Jannetta
Singular and nonsingular matrices The determinant provides important information about the matrix. A matrix whose determinant is zero is said to be singular. A matrix whose determinant is not zero is said to be nonsingular. For example, the equations 2x+3y = 10 4x 5y = 16 can be represented in matrix form by 2 3 x 4 5 y = 10 16 The determinant of the matrix of coefficients tells us whether or not there are solutions to original set of equations. Here, we have A= 2 3 4 5 det A= 22 Since A is nonsingular then the two equations do have solutions. If A had been singular (deta = 0) then it would not have solutions. Determinants, minors and cofactors 13/ 15 Adrian Jannetta We will soon be using matrices to solve systems of equations and the
Solutions of linear systems Determine whether or not the equations 2.5x+13y = 7.25 35x 182y = 101.5 have a unique solution. In matrix form the equations are Ax=b: 2.5 13 x 35 182 y = 7.25 101.5 The matrix of coefficients will tell us about the solution the RHS of the equation is unimportant. 2.5 13 A= 35 182 The determinant is det A=( 2.5)( 182) (13)(35)= 455 455=0 Therefore, there are no unique solutions to the original equations. Determinants, minors and cofactors 14/ 15 Adrian Jannetta
Test yourself If you ve understood the ideas and examples presented in these notes then you should be able to solve the following problems. 2 1 3 4 2 12 8 Given the matrices A=, B= and C= 4 1 4. 7 3 6 4 5 0 8 1 Evaluate det A. 2 Show that B is a singular matrix. 3 Find detc. 4 Write down the cofactor matrix for C. 1 det A=26. 2 det B = 0, therefore B is singular. 3 det C=83. 4 Cofactor matrix 8 52 5 8 31 5 7 4 6 Determinants, minors and cofactors 15/ 15 Adrian Jannetta