Linear Algebra

300 Chapter Four. Determinantš 1.11 Many people know this mnemonic for the determinant of a 3×3 matrix: firstrepeat the first two columns and then sum the products on the forward diagonalsand subtract the products on the backward diagonals. That is, first write⎛⎞h 1,1 h 1,2 h 1,3 h 1,1 h 1,2⎝h 2,1 h 2,2 h 2,3 h 2,1 h 2,2⎠h 3,1 h 3,2 h 3,3 h 3,1 h 3,2and then calculate this.h 1,1 h 2,2 h 3,3 + h 1,2 h 2,3 h 3,1 + h 1,3 h 2,1 h 3,2−h 3,1 h 2,2 h 1,3 − h 3,2 h 2,3 h 1,1 − h 3,3 h 2,1 h 1,2(a) Check that this agrees with the formula given in the preamble to this section.(b) Does it extend to other-sized determinants?1.12 The cross product of the vectors⎛ ⎞ ⎛ ⎞x 1y 1⃗x = ⎝x 2⎠ ⃗y = ⎝y 2⎠x 3 y 3is the vector computed as this determinant.⎛⎞⃗e 1 ⃗e 2 ⃗e 3⃗x × ⃗y = det( ⎝x 1 x 2 x 3⎠)y 1 y 2 y 3Note that the first row’s entries are vectors, the vectors from the standard basis forR 3 . Show that the cross product of two vectors is perpendicular to each vector.1.13 Prove that each statement holds for 2×2 matrices.(a) The determinant of a product is the product of the determinants det(ST) =det(S) · det(T).(b) If T is invertible then the determinant of the inverse is the inverse of thedeterminant det(T −1 ) = ( det(T) ) −1 .Matrices T and T ′ are similar if there is a nonsingular matrix P such that T ′ = PTP −1 .(This definition is in Chapter Five.) Show that similar 2×2 matrices have the samedeterminant.̌ 1.14 Prove that the area of this region in the plane( )x2y 2( )x1y 1is equal to the value of this determinant.( )x1 x 2det( )y 1 y 2Compare with this.( )x2 x 1det( )y 2 y 11.15 Prove that for 2×2 matrices, the determinant of a matrix equals the determinantof its transpose. Does that also hold for 3×3 matrices?̌ 1.16 Is the determinant function linear — is det(x · T + y · S) = x · det(T) + y · det(S)?1.17 Show that if A is 3×3 then det(c · A) = c 3 · det(A) for any scalar c.