MATH 125 Lecture Notes - Lecture 31: Elementary Matrix, Triangular Matrix, Laplace Expansion

Let A be a 4 times 4 matrix and B be obtained from A by adding 5 times the first row to each of the second and third rows, then det (B) = 5^4 det (A) True False (h) A square matrix D is invertible if and only if 1/det(D) exists as a real number True False (i) If a square matrix B is invertible, then its inverse has zero determinant. True False (j) In order to apply Cramer's Rule, the coefficient matrix of a system must be invertible and the determinant of the inverse of the coefficient matrix must be different from 0. True False

Show that the set (U,,U2 Ug ) is an orthogonal basis for R.Then express the vector X as a ar combinations of the vectors U's. Where 0 15. Let A and B are n x n matrices. Mark each statement True or False, Justify your answer (a) The determinant of a matrix A is the product of the diagonal entries in A. (b) (det A) (det B) det AB (c) det A (-1) det A (d) Similar matrices always have exactly the same eigen values (e) If A is invertible, then the equation AX b is consistent for each b in R.

Express each of the following 2D geometric transformations as a matrix, and explain how you would define the reverse transformation (inverse). Use homogeneous coordinates when necessary. Uniform scale with respect to the origin Rotation by theta degrees about the origin in the counterclockwise direction Translation by an amount (tx, ty) Let A = Using hand calculations, compute the determinant of A using co-factor expansions. At each step, choose the row or column that involves the least amount of work. You can check your answer using Matlab. Let A = ,and let k be a scalar. Find a formula that relates the determinant of A to the determinant of kA. If A were a 2 Times 2 matrix, what would the formula be? Generalize to the case of an n Times n matrix. Without using Matlab, compute Compare det(A) with each of the following: det(A'), det(-A) and det(A-1) for simple square matrices of various sizes and make conjectures about how each of these determinants are related. Provide a printout showing your work. You may find it helpful to use the format command in Matlab to output some of your results in rational form. When A is an arbitrary m Times n matrix that has more columns that rows, what do you know about the matrix products AT A and AAT? What do you know about det(ATA)? Explain your answer.