 6.3.1: Find the area of the parallelogram defined by8 2
 6.3.2: ind the area of the triangle defined by 3" 8' and 7 2_
 6.3.3: Find the area of the following triangle:
 6.3.4: Consider the area A of the triangle with vertices . Express A in te...
 6.3.5: The tetrahedron defined by three vectors 5i, 52,53 in R3 is the set...
 6.3.6: What is the relationship between the volume of the tetrahedron defi...
 6.3.7: Find the area of the following region:
 6.3.8: Demonstrate the equation  det ,4 =for [5,1*2 I W,na noninvertible...
 6.3.9: If 51 and v2 are linearly independent vectors in R2, what is the re...
 6.3.10: Consider an nxn matrix A = [5i V 2 5]. What is the relationship bet...
 6.3.11: Consider a linear transformation T(x) = Ax from R2 to R2. Suppose f...
 6.3.12: Consider those 4 x 4 matrices whose entries are all 1, 1, or 0. Wha...
 6.3.13: Find the area (or 2volume) of the parallelogram (or 2 parallelepi...
 6.3.14: Find the 3volume of the 3parallelepiped defined by the vectors
 6.3.15: Demonstrate Theorem 6.3.6 for linearly dependent vectors Si,..., 5m.
 6.3.16: True or false? If Q is a parallelogram in R3 and T (jc) = Ax is a l...
 6.3.17: (For some background on the cross product in Rw, see Exercise 6.2.4...
 6.3.18: If T (Jc) = Ax is an invertible linear transformation from R2 to R2...
 6.3.19: A basis Si, 52, S3 of R3 is called positively oriented if Si enclos...
 6.3.20: We say that a linear transformation T from R3 to R3 preserves orien...
 6.3.21: Arguing geometrically, determine whether the following orthogonal t...
 6.3.22: Use Cramers rule to solve the systems in Exercises 22 through 24.
 6.3.23: Use Cramers rule to solve the systems in Exercises 22 through 24.
 6.3.24: Use Cramers rule to solve the systems in Exercises 22 through 24.
 6.3.25: Find the classical adjoint of the matrixA =1 0 1 0 1 0 2 0 1and use...
 6.3.26: Consider an nxn matrix A with integer entries such that det A = 1. ...
 6.3.27: Consider two positive numbers a and b. Solve the following system:a...
 6.3.28: In an economics text,10 we find the following system:sY +ar = / c +...
 6.3.29: In an economics text11 we find the following system:R i *i a 1 a R...
 6.3.30: Find the classical adjoint of A =
 6.3.31: Find the classical adjoint of A =
 6.3.32: Find the classical adjoint of A =
 6.3.33: Find the classical adjoint of A =
 6.3.34: For an invertible n x n matrix A, find the product A(adjA). What ab...
 6.3.35: For an invertible nxn matrix A, what is the relationship between de...
 6.3.36: For an invertible nxn matrix A, what is adj (adj A)?
 6.3.37: For an invertible nxn matrix A, what is the relationship between ad...
 6.3.38: For two invertible nxn matrices A and B, what is the relationship b...
 6.3.39: If A and B are invertible n x n matrices, and if A is similar to B,...
 6.3.40: For an invertible nxn matrix A, consider the linear transformationT...
 6.3.41: Show that an nxn matrix A has at least one nonzero minor if (and on...
 6.3.42: Even if an n x n matrix A fails to be invertible, we can define the...
 6.3.43: Show that A (adj A ) = 0 = (adj A) A for all noninvertible nxn matr...
 6.3.44: If A is an n x n matrix of rank n 1, what is the rank of adj (A)? S...
 6.3.45: Find all 2 x 2 matrices A such that adj (A) = AT.
 6.3.46: (For those who have studied multivariable calculus.) Let T be an in...
 6.3.47: Consider the quadrilateral in the accompanying figure, with vertice...
 6.3.48: What is the area of the largest ellipse you can inscribe into a tri...
 6.3.49: What are the lengths of the semiaxes of the largest ellipse you can...
Solutions for Chapter 6.3: Geometrical Interpretations of the Determinant; Cramers Rule
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 6.3: Geometrical Interpretations of the Determinant; Cramers Rule
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Chapter 6.3: Geometrical Interpretations of the Determinant; Cramers Rule includes 49 full stepbystep solutions. Since 49 problems in chapter 6.3: Geometrical Interpretations of the Determinant; Cramers Rule have been answered, more than 14550 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.