 45.1: 7 3 8 2
 45.2: 4 6 8
 45.3: 0 3 2 4 2 1 0 5 1
 45.4: 2 6 1 3 5 2 4 7 8
 45.5: 1 2 1 6 3 6 4 1 4
 45.6: 1 3 3 4 2 1 0 5 2
 45.7: GEOMETRY What is the area of ABC with A(5, 4), B(3, 4), and C(3, ...
 45.8: Find the area of the triangle whose vertices are located at (2, 1)...
 45.9: 10 5 6 5
 45.10: 8 6 5 1
 45.11: 9 3 7
 45.12: 2 3 4 6
 45.13: 6 8 2 5
 45.14: 9 12 0 7
 45.15: 4 5.2 1.6
 45.16:  3.2 4.1 5.8 3.9
 45.17: 3 0 2 1 6 5 2 4 1
 45.18: 7 2 0 3 9 0 4 6 0
 45.19: 2 4 1 7 5 0 2 2 1
 45.20: 3 6 1 0 5 4 6 2 2
 45.21: 1 7 6 5 3 3 4 2 1
 45.22: 3 1 8 7 6 3 6 2 5
 45.23: 1 3 8 1 9 7 1 5 4
 45.24: 1 6 5 5 7 9 2 8 3
 45.25: 8 1 6 9 5 2 0 4 3
 45.26: GEOGRAPHY Mr. Cardona is a regional sales manager for a company in ...
 45.27: ARCHAEOLOGY During an archaeological dig, a coordinate grid is laid...
 45.28: GEOMETRY Find the area of a triangle whose vertices are located at ...
 45.29: GEOMETRY Find the area of the polygon shown at the right.
 45.30: Solve for x if det 2 5 x 3 = 24.
 45.31: Solve det 4 x 6 x 3 2 2 1 3 = 3 for x.
 45.32: GEOMETRY Find the value of x such that the area of a triangle whose...
 45.33: GEOMETRY The area of a triangle ABC is 2 square units. The vertices...
 45.34: 3 8  6.5 3.75
 45.35: 10 40 70 20 50 80 30 60 90
 45.36: 10 3 16 12 18 2 4 9 1
 45.37: OPEN ENDED Write a matrix whose determinant is zero.
 45.38: FIND THE ERROR Khalid and Erica are finding the determinant of 8 5...
 45.39: REASONING Find a counterexample to disprove the following statement...
 45.40: CHALLENGE Find a thirdorder determinant in which no element is 0, ...
 45.41: Writing in Math Use the information about the Bermuda Triangle on p...
 45.42: ACT/SAT Find the area of triangle ABC. A 10 units 2 B 12 un its 2 C...
 45.43: REVIEW Use the table to determine the expression that best represen...
 45.44: Write the coordinates of ABC in a vertex matrix. 45. Find the coord...
 45.45: Find the coordinates of A B C . Then graph ABC and A B C .
 45.46: 2 2 4 3 3 1 9 2
 45.47: 5 7 1 4 6 2
 45.48: 7 6 5 1 4 3 1 2 1 3 8 2
 45.49: MARATHONS The length of a marathon was determined in the 1908 Olymp...
 45.50: slope 1, passes through (5, 3)
 45.51: slope _4 3 , passes through (6, 8)
 45.52: passes through (3, 7) and (2, 3)
 45.53: passes through (0, 5) and (10, 10)
 45.54: x + y = 3 3x + 4y = 12
 45.55: x + y = 10 2x + y = 11
 45.56: 2x + y = 5 4x + y = 9
Solutions for Chapter 45: Determinants
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 45: Determinants
Get Full SolutionsAlgebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Chapter 45: Determinants includes 56 full stepbystep solutions. Since 56 problems in chapter 45: Determinants have been answered, more than 55638 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).