 3.1.1: In Eercises 1 8, compute the dererminam of each matrix. ~ 
 3.1.2: In Eercises 1 8, compute the dererminam of each matrix. 2. ~ ~
 3.1.3: In Eercises 1 8, compute the dererminam of each matrix. 3. ~ ~]
 3.1.4: In Eercises 1 8, compute the dererminam of each matrix. 4. ~
 3.1.5: In Eercises 1 8, compute the dererminam of each matrix. 5. [5 10 6
 3.1.6: In Eercises 1 8, compute the dererminam of each matrix. 6. [~ ~J
 3.1.7: In Eercises 1 8, compute the dererminam of each matrix. 7. ~ n
 3.1.8: In Eercises 1 8, compute the dererminam of each matrix. 8. [~
 3.1.9: In Exercises 9 12. compute each indicated cofactor of the matrix A...
 3.1.10: In Exercises 9 12. compute each indicated cofactor of the matrix A...
 3.1.11: In Exercises 9 12. compute each indicated cofactor of the matrix A...
 3.1.12: In Exercises 9 12. compute each indicated cofactor of the matrix A...
 3.1.13: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.14: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.15: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.16: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.17: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.18: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.19: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.20: In Exercises 1320, compute the determinam of each matrix A by a co...
 3.1.21: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.22: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.23: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.24: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.25: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.26: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.27: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.28: In Exercises 2128, compwe each determina/11 by any legitimate meth...
 3.1.29: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.30: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.31: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.32: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.33: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.34: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.35: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.36: In Exercises 2936. compute the area of each parallelogram determin...
 3.1.37: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.38: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.39: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.40: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.41: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.42: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.43: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.44: In Exercises 3744, find each value of c for which the matrix is no...
 3.1.45: The determinant of a matrix is a matrix of the same size
 3.1.46: de [~ !] =ad+
 3.1.47: If the determinant of a 2 x 2 matrix etjuals zero. then the matrix ...
 3.1.48: If a 2 x 2 matrix is invertible. then its determinant equals zero.
 3.1.49: If 8 is a matrix obtained by multiplying each entry of some row of ...
 3.1.50: For n ;:: 2, the (i ,j)cofactor of an 11 x 11 matrix A is the dete...
 3.1.51: For 11 2: 2. the (i .j)cofactor of an n x n matrix A equals ( 1) ...
 3.1.52: The dete1minant of an n x n matrix can be evaluated by a cofactor e...
 3.1.53: Cofactor expansion is an efficient method for evaluati ng the deter...
 3.1.54: The determinant of a matrix with integer entries must be ;m integer
 3.1.55: The detenninant of a matrix with po.otive entries must be positive
 3.1.56: If ome row of a 'quare matrix con'i't~ only of Lero entrie,, then t...
 3.1.57: An upper triangular matrix must be square.
 3.1.58: A matrix in which a ll the e ntries to the left :md below the di:go...
 3.1.59: A 4 x 4 upper triangular matrix has at mo't 10 nonzero entries
 3.1.60: The tran,pose of a lower triangular matrix i' an upper triangular m...
 3.1.61: The determinant of an upper triangular 11 x 11 matrix or a lo"er tr...
 3.1.62: The detenninant of In equals I.
 3.1.63: 1l1e :trea of the parallelogram determined by u and ,, is det [u ' '].
 3.1.64: If T: 'R.2  'R.2 is a linear transformation, then del [T(u) T(v)] ...
 3.1.65: Show that the determinant of the rotation matrix ~ is 1.
 3.1.66: Show that if A is a 2 x 2 matrix 111 "hich every entry is 0 or I. t...
 3.1.67: Show that the conclusion of Exercise 66 is false for 3 x 3 matrices...
 3.1.68: Prove that if a 2 x 2 matrix has identical rows, then its determina...
 3.1.69: Prove that, for any 2 x 2 matrix A, dctAr =
 3.1.70: Let A be a 2 x 2 matrix and k be a scalar. llow does detkA compare ...
 3.1.71: Prove that, for any 2 x 2 matrices A and 8. det AB = (detAXdet 8).
 3.1.72: What is the determinant of an 11 x 11 matrix with a zero row? Justi...
 3.1.73: For t'ach t'lmumaf') matrix E in E.urr:ISt'S 73 76. and for tht' mt...
 3.1.74: For t'ach t'lmumaf') matrix E in E.urr:ISt'S 73 76. and for tht' mt...
 3.1.75: For t'ach t'lmumaf') matrix E in E.urr:ISt'S 73 76. and for tht' mt...
 3.1.76: For t'ach t'lmumaf') matrix E in E.urr:ISt'S 73 76. and for tht' mt...
 3.1.77: Prove that det[ a c +~p b J =det[a "+ kt/ ,. ~
 3.1.78: The Tl85 calculator gives del[~ 2 3 4 n = 4 X 10 13 Why must thi...
 3.1.79: Ue a determmant to express the area of the triangle 111 'R.2 havin...
 3.1.80: Calculate the determinant of [~ ; .] if 0 and 0 ' are zero matrice>.
 3.1.81: (a) Gcncr:uc random 4 x 4 matrices A and 8 . Evaluate detA, det8. a...
 3.1.82: (a) Generate r.tndom 4 x l matrices A and 8 . Evaluate detA, del B...
 3.1.83: (a) Gcncmtc a random 4 x 4 matrix A. Evaluate dct A and dctA'. (b) ...
 3.1.84: a) Let A= [ ! 1 I I 2 0 0 2 ~] I . I Show that A is invertible...
Solutions for Chapter 3.1: DETERMINANTS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 3.1: DETERMINANTS
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Elementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. This expansive textbook survival guide covers the following chapters and their solutions. Since 84 problems in chapter 3.1: DETERMINANTS have been answered, more than 34993 students have viewed full stepbystep solutions from this chapter. Chapter 3.1: DETERMINANTS includes 84 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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 .

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.