Linear Algebra

Section I. Definition 291Exerciseš 1.1 Evaluate the determinant of each.( ) ( )2 0 13 1(a)(b) 3 1 1−1 1−1 0 11.2 Evaluate the determinant of each.( ) ( )2 1 12 0(a)(b) 0 5 −2−1 31 −3 4(c)(c)( 4 0) 10 0 11 3 −1( 2 3) 45 6 78 9 1̌ 1.3 Verify that the determinant of an upper-triangular 3×3 matrix is the productdown the diagonal.( ) a b cdet( 0 e f ) = aei0 0 iDo lower-triangular matrices work the same way?̌ 1.4 Use( the determinant ) ( to decide ) if each ( is singular ) or nonsingular.2 10 14 2(a)(b)(c)3 11 −12 11.5 Singular or nonsingular? Use the determinant to decide.( ) ( ) ( )2 1 11 0 12 1 0(a) 3 2 2 (b) 2 1 1 (c) 3 −2 00 1 44 1 31 0 0̌ 1.6 Each pair of matrices differ by one row operation. Use this operation to comparedet(A) with ( det(B). ) ( )1 2 1 2(a) A = B =(b) A =(c) A =2 3( 3 1 00 0 10 1 2)( 1 −1 32 2 −61 0 4B =)0 −1( 3 1) 00 1 20 0 1( )1 −1 3B = 1 1 −31 0 41.7 Show this. ( )1 1 1det( a b c ) = (b − a)(c − a)(c − b)a 2 b 2 c 2̌ 1.8 Which real numbers x make(this matrix singular?)12 − x 48 8 − x1.9 Do the Gaussian reduction to check the formula for 3×3 matrices stated in thepreamble to this section.( ) a b cd e f is nonsingular iff aei + bfg + cdh − hfa − idb − gec ≠ 0g h i1.10 Show that the equation of a line in R 2 thru (x 1, y 1) and (x 2, y 2) is expressedby this determinant.( )x y 1det(x 1 y 1 1x 2 y 2 1) = 0 x 1 ≠ x 2