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Lecture 28.pdf

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MATH 136
Robert Sproule

Monday, March 17 − Lecture 28 : Determinants Concepts: 1. Define the (i, j)-minor of a square matrix. 2. Define the (i, j)-cofactor of a square matrix. 3. Calculate the determinant of a square matrix by row or column expansion. 4. Apply the fact that the determinant of a triangular matrix is the product of the entries on the diagonal. 5. Recognize that a triangular matrix A is invertible iff its determinant is not 0. 28.1 Determinants − Let M denote the set of all square matrices. A “determinant” can be viewed as a function D : M → ℝ which maps any n × n matrix A to a real number according to well defined formula applied to the entries of the matrix. Its general formula will be stated explicitly below. We denote the determinant function by det(A) or | A |. In the case of a 2 × 2 matrix A = [a ] ij 2 × 2the function “det” is defined as: det(A) = a a 11 22 . 12 21 28.2 Definition − Let A = [a ] be an ij× n matrix. For each entry a in A, we associateij matrix M of ijmension n − 1 by n − 1 whose entries are the ones that remain when we remove from A the row and the column containing the entry a . We will refer to ijas a ij minor matrix of A associated to a ij We will call the number m ijdet(M ) = |ij | ij the minor of the elementa , or the (i, j) minor. ij Remark − Note that M is a matijx while m is a real numbeij Also note that we have not yet defined what det(M ). ij Second Remark − For an n × n matrix A there are n minors m , one for each entryijf A. 28.3 Definition − For each minor m of an n × nijatrix A, we define the (i, j)-cofactor c as ij i + j c ij(−1) m .ij 28.3.1 Example. Let Abe the matrix Then (2, 3)-minor matrix, M , is 23 2 + 3 Then m =23et(M ) = 23– 0 = 3 and c = (–1)23 m 23–3. 28.4 Definition − For an n × n matrix A = [a ] ij n × nSuppose the det(B m × m) is defined for any positive integer m < n. Then all (i, j)-cofactors of A are all well-defined numbers. We define the determinant of A, det(A) = | A | as : Remark – Let M denote the set of all square matrices and M denote the set of all n × n n × n matrices for some n. According to this definition we see that the determinant function D : M → ℝ is a “recursively defined function” on the set of positive integers. This means that we cannot determine the values of the determinants D: M n × n → ℝ unless we know the values of the determinants in D : M m × m→ ℝ for all m < n. It can be proven that such recursively defined functions are well-defined functions on the set of all square matrices M. Having acknowledged this, we will henceforth accept this as a proven fact. Remark – We will soon see that there are many(often more convenient) ways of computing the value of a determinant det(A). 28.4.1 Example – Let A be the matrix We begin the “cofactor expansion along row 1” allowing us to find det A: We compute det M : 14 So We allow the reader to verify in this way the values of det M 11, det M 12d det M . 13 By proceeding in this ways for the other minors we get det(A) = −256. 28.4.2 Remark − This method of evaluating a determinant of A is referred as “the cofactor expansion of det(A) along the first row”. 28.5 Theorem − For a square n × n matrix A = [a ] ij n × ne determinant det(A) can also be computed by th (e
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