Kenneth Kuttler

500 CHAPTER 27. DETERMINANTS

Example 27.1.11 Find det(A) where

A =

1 2 3 45 4 2 31 3 4 53 4 3 2

As in the case of a 3× 3 matrix, you can expand this along any row or column. Lets

pick the third column. det(A) =

3(−1)1+3

∣∣∣∣∣∣∣5 4 31 3 53 4 2

∣∣∣∣∣∣∣+2(−1)2+3

∣∣∣∣∣∣∣1 2 41 3 53 4 2

∣∣∣∣∣∣∣+4(−1)3+3

∣∣∣∣∣∣∣1 2 45 4 33 4 2

∣∣∣∣∣∣∣+3(−1)4+3

∣∣∣∣∣∣∣1 2 45 4 31 3 5

∣∣∣∣∣∣∣ .Now you know how to expand each of these 3×3 matrices along a row or a column. If youdo so, you will get −12 assuming you make no mistakes. You could expand this matrixalong any row or any column and assuming you make no mistakes, you will always getthe same thing which is defined to be the determinant of the matrix A. This method ofevaluating a determinant by expanding along a row or a column is called the method ofLaplace expansion.

Note that each of the four terms above involves three terms consisting of determinantsof 2×2 matrices and each of these will need 2 terms. Therefore, there will be 4×3×2= 24terms to evaluate in order to find the determinant using the method of Laplace expansion.Suppose now you have a 10× 10 matrix and you follow the above pattern for evaluatingdeterminants. By analogy to the above, there will be 10! = 3,628 ,800 terms involved inthe evaluation of such a determinant by Laplace expansion along a row or column. This isa lot of terms.

In addition to the difficulties just discussed, you should regard the above claim that youalways get the same answer by picking any row or column with considerable skepticism. Itis incredible and not at all obvious. However, it requires a little effort to establish it. Thisis done in the section on the theory of the determinant.

Definition 27.1.12 Let A = (ai j) be an n× n matrix and suppose the determinant of a(n−1)× (n−1) matrix has been defined. Then a new matrix called the cofactor ma-trix, cof(A) is defined by cof(A) = (ci j) where to obtain ci j delete the ith row and the jth

column of A, take the determinant of the (n−1)× (n−1) matrix which results, (This iscalled the i jth minor of A. ) and then multiply this number by (−1)i+ j. Thus (−1)i+ j×(the i jth minor

)equals the i jth cofactor. To make the formulas easier to remember, cof(A)i j

will denote the i jth entry of the cofactor matrix.

With this definition of the cofactor matrix, here is how to define the determinant of ann×n matrix.

500 CHAPTER 27. DETERMINANTSExample 27.1.11 Find det (A) whereBDO = neRW BWVBO BN wWYN nw sfAs in the case of a 3 x 3 matrix, you can expand this along any row or column. Letspick the third column. det (A) =5 4 3 12 43(-1)3f 1305 f42(-1)8] 1353.4 2 3.4 212 4 12 4+4(-1)79] 5 4 3 143(-1)°°%] 5 4 33 4 2 13 5Now you know how to expand each of these 3 x 3 matrices along a row or a column. If youdo so, you will get —12 assuming you make no mistakes. You could expand this matrixalong any row or any column and assuming you make no mistakes, you will always getthe same thing which is defined to be the determinant of the matrix A. This method ofevaluating a determinant by expanding along a row or a column is called the method ofLaplace expansion.Note that each of the four terms above involves three terms consisting of determinantsof 2 x 2 matrices and each of these will need 2 terms. Therefore, there will be 4 x 3 x 2 = 24terms to evaluate in order to find the determinant using the method of Laplace expansion.Suppose now you have a 10 x 10 matrix and you follow the above pattern for evaluatingdeterminants. By analogy to the above, there will be 10! = 3,628 , 800 terms involved inthe evaluation of such a determinant by Laplace expansion along a row or column. This isa lot of terms.In addition to the difficulties just discussed, you should regard the above claim that youalways get the same answer by picking any row or column with considerable skepticism. Itis incredible and not at all obvious. However, it requires a little effort to establish it. Thisis done in the section on the theory of the determinant.Definition 27.1.12 Let A = (ajj) be an nxn matrix and suppose the determinant of a(n—1) x (n—1) matrix has been defined. Then a new matrix called the cofactor ma-trix, cof (A) is defined by cof (A) = (ci) where to obtain c;; delete the i" row and the j'"column of A, take the determinant of the (n—1) x (n—1) matrix which results, (This iscalled the ij'" minor of A. ) and then multiply this number by (—1)'*!. Thus (—1)'*! x(the ijt minor) equals the ij'" cofactor. To make the formulas easier to remember, cof (A); jwill denote the ij'" entry of the cofactor matrix.With this definition of the cofactor matrix, here is how to define the determinant of ann Xn matrix.