Unformatted text preview: Solutions to problems from section 3.1 2. Compute the determinant of the following matrix using the cofactor expansion across the ﬁrst row and then compute the determinant by a cofactor expansion down the second column. 0 51 4 −3 0 2 41 Solution: 0 51 4 −3 0 2 41 0 51 4 −3 0 41 2 =0 −3 0 41 −5 40 21 +1 4 −3 2 4 = −5(4) + 1(22) = 2. = −5 40 21 −3 01 21 −4 01 40 = −5(4) − 3(−2) − 4(−4) = 2. 4. Compute the determinant of the following matrix using the cofactor expansion across the ﬁrst row and then compute the determinant by a cofactor expansion down the second column. 135 2 1 1 342 Solution: 135 211 342 135 211 342 =1 11 42 −3 21 32 +5 21 34 = 1(−2) − 3(1) + 5(5) = 20. = −3 21 32 +1 15 32 −4 15 21 = −3(1) + 1(−13) − 4(−9) = 20. 10. Compute the determinant by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. 1 −2 5 0 0 3 2 −6 −7 5 0 4 Solution: 1 −2 5 0 0 3 2 −6 −7 5 0 4 2 0 5 4 1 −2 2 = −3 2 −6 5 5 04 −2 2 −6 5 1 −2 2 −6 2 0 5 4 = −3 5 +4 = −3(5(2)+4(−2)) = −6. 12. Compute the determinant by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. 4 00 0 7 −1 0 0 2 63 0 5 −8 4 −3 Solution: 4 00 0 7 −1 0 0 2 63 0 5 −8 4 −3 −1 0 0 =4 63 0 −8 4 −3 3 0 4 −3 = 4(−1) = 4(−1)(3)(−3) = 36. abc 322 24. State the row operation which takes 3 2 2 to a b c and describe how it aﬀects 656 656 the determinant. Solution: The row operation is R1 ↔ R2 . Notice that the determinant of the original matrix is the negative of the determinant of the matrix obtained after the row operation. Indeed, abc 322 656 = − −a =a 22 56 −b 32 66 +c 32 65 22 56 +b 32 66 −c 32 65 322 =− a b c 656 38. Let A = ab cd and let k be a scalar. Find a formula relating det A and k to det(kA). Solution: Notice that det kA = ka kb kc kd = (ka)(kd) − (kb)(kc) = k2 (ad − bc) = k2 det A. Thus det kA = k2 det A 40. True or false. Justify your answer. a. The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row. Solution: FALSE. They are the same. b. The determinant of a triangular matrix is the sum of the entries on the main diagonal. Solution: FALSE. The determinant of a triangular matrix is the product of the entries on the main diagonal. For example det I2 = 1 = 2. 42. Let u = a c and v = , where a, b, c are positive (for simplicity). Compute the area b 0 of the parallelogram determined by u, v, u + v, and 0, and compute the determinants of the matrices [ u v ] and [ v u ]. Draw a picture and explain what you ﬁnd. Solution: Well, here’s a picture: To compute the area of the parallelogram we need to ﬁnd know the length of its base and its height of the parallelogram. From the picture we see that the base has length c and the height of the parallelogram is b. Thus the area of the parallelogram is bc Notice det[ u v ] = and det[ v u ] = ca 0b = bc. ac b0 = −bc So we see that the area of the parallelogram is equal to det[ v u ] = − det[ u v ]. ...
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 Fall '08
 AXENOVICH
 Determinant, Harshad number, Characteristic polynomial, Diagonal matrix

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