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Math 324: Linear Algebra Section 3.1: the Determinant of a Matrix

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Math 324: Linear Algebra Section 3.1: the Determinant of a Matrix Math 324: Linear Algebra Section 3.1: The Determinant of a Matrix Mckenzie West Last Updated: March 3, 2020 2 Last Time. { Proof by Induction Today. { The determinant of a 2 × 2 matrix { Minors { Cofactors { The determinant of an n × n matrix. 3 Definition. Generally speaking, the determinant of an n × n matrix is a real number that we can associate to it that indicates whether A~x = ~0 has nontrivial solutions. Note. Recall that A~x = ~0 has a unique solution if and only if A is invertible. Therefore, the determinant can also indicate whether A is invertible. Question 1. In light of this discussion, what do you think the determinant of a a the matrix A = 11 12 is? a21 a22 4 Notation. If A is an 2 × 2 matrix, denote the determinant of A by jAj, det(A), a11 a12 or = a11a22 − a12a21: a21 a22 Exercise 2. Compute each of the following: 1 −5 2 1 − 0 0 (a) (b) 3 (c) 4 −3 −2 −5 1 −1 Question 3. What are the possible values for jAj? 5 Definition. If A is a square matrix, then the( i; j)-minor, Mij , is the determinant of the matrix obtained by deleting the ith row and jth column of A. Exercise 4. Determine the given minor: 2 0 3 33 (a) (1; 2)-minor of 4 4 3 55 −1 0 3 2−1 2 4 3 (b) (3; 3)-minor of 4 2 −1 −35 1 4 5 2 3 a11 a12 a13 (c) (2; 1)-minor of 4a21 a22 a235 a31 a32 a33 6 Definition. If A is a square matrix, then the( i; j)-cofactor, Cij , is a signed i+j version of the (i; j)-minor given by Cij = (−1) Mij . Exercise 5. Determine the given cofactor: 2 0 3 33 (a) (1; 2)-cofactor of 4 4 3 55 −1 0 3 2−1 2 4 3 (b) (3; 3)-cofactor of 4 2 −1 −35 1 4 5 2 3 a11 a12 a13 (c) (2; 1)-cofactor of 4a21 a22 a235 a31 a32 a33 7 Note. The sign of the cofactor alternates across rows and columns: 2+ − + − + ···3 2+ − + −3 6− + − + − · · ·7 2+ − +3 6 7 − + − + 6+ − + − + ···7 − + − 6 7 6 7 4 5 6+ − + −7 6− + − + − · · ·7 + − + 4 5 6 7 − + − + 6+ − + − + ···7 4 . 5 . 3 × 3 4 × 4 n × n 8 Definition. If A is an n × n matrix (n ≥ 2), the determinant of A is the value n X det(A) = jAj = a11C11 + a12C12 + ··· + a1nC1n = a1j C1j : j=1 Note. We call this an inductive definition because we use the determinant of one size smaller matrix to define the determinant of the next. Exercise 6. Use the definition given here to compute 0 3 3 4 3 5 −1 0 3 9 Brain Break. What band were you obsessed with in middle school? 10 Note. Notice how convenient it was that one of the entries in the first row of the last matrix was 0. We didn't actually have to compute the (1; 1)-cofactor to compute the determinant. We can actually take advantage of rows (and columns!) that have a lot of zeros when we take determinants. 11 Theorem 3.1(Expansion by Cofactors) Let A be a square matrix of order n. Then the determinant of A is given by: (a) the i-th row expansion n X det(A) = jAj = ai1Ci1 + ai2Ci2 + ··· + ainCin = aij Cij : j=1 (b) or the j-th column expansion n X det(A) = jAj = c1j C1j + a2j C2j + ··· + anj Cnj = aij Cij : i=1 12 Exercise 7. Compute the determinant of the following matrix in two different ways: (a) the 3rd row expansion and (b) the 4th column expansion: 2 −4 −2 5 0 3 6 −3 −1 0 0 7 6 7 : 4 0 0 −5 0 5 −5 0 0 3 Hopefully you got the same answer in both cases. Why do you think that happened? 13 Exercise 8. Use your choice of row/column cofactor expansion(s) to compute each of the following: −2 1 0 −5 5 0 −4 (a) 4 0 −2 5 4 0 0 (c) −4 −1 5 −1 −2 2 −3 −4 0 0 2 2 2 −2 1 0 −1 5 −2 1 −3 −2 0 (b) 0 0 −3 3 −3 0 1 1 (d) 0 0 0 4 5 0 0 0 −5 0 −4 −4 14 Exercise 9. Determine the values of λ for which the determinant of the given matrix is zero: λ + 2 0 (a) −5 λ − 3 λ 1 (b) 7 λ + 6 2λ + 1 0 03 (c) 4 1 λ − 2 15 1 0 λ 15 Exercise 10. Evaluate the determinant to verify the equation: a b c d (a) = − c d a b 2 1 x x (b) 1 y y 2 = (y − x)(z − x)(z − y) 1 z z2 16 Exercise 11. Show that the system of linear equations ax + by = e cx + dy = f has a unique solution if and only if the determinant of the coefficient matrix is nonzero..
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