 4.5.1: Find the value of each determinant.
 4.5.2: Find the value of each determinant.
 4.5.3: Evaluate each determinant using expansion by minors.
 4.5.4: Evaluate each determinant using expansion by minors.
 4.5.5: Evaluate each determinant using diagonals
 4.5.6: Evaluate each determinant using diagonals
 4.5.7: Evaluate each determinant using diagonals
 4.5.8: Find the area of the triangle whose vertices are located at (2, 1)...
 4.5.9: Find the value of each determinant.
 4.5.10: Find the value of each determinant.
 4.5.11: Find the value of each determinant.
 4.5.12: Find the value of each determinant.
 4.5.13: Find the value of each determinant.
 4.5.14: Find the value of each determinant.
 4.5.15: Find the value of each determinant.
 4.5.16: Find the value of each determinant.
 4.5.17: Find the value of each determinant.
 4.5.18: Find the value of each determinant.
 4.5.19: Find the value of each determinant.
 4.5.20: Find the value of each determinant.
 4.5.21: Find the value of each determinant.
 4.5.22: Find the value of each determinant.
 4.5.23: Find the value of each determinant.
 4.5.24: Find the value of each determinant.
 4.5.25: Find the value of each determinant.
 4.5.26: Mr. Cardona is a regional sales manager for a company in Florida. T...
 4.5.27: During an archaeological dig, a coordinate grid is laid over the si...
 4.5.28: Find the area of a triangle whose vertices are located at (4, 1), (...
 4.5.29: Find the area of the polygon shown at the right.
 4.5.30: Solve for x if det 2 5 x 3 = 24.
 4.5.31: Solve det 4 x 6 x 3 2 2 1 3 = 3 for x.
 4.5.32: Find the value of x such that the area of a triangle whose vertices...
 4.5.33: The area of a triangle ABC is 2 square units. The vertices of the t...
 4.5.34: Use a graphing calculator to find the value of each determinant.
 4.5.35: Use a graphing calculator to find the value of each determinant.
 4.5.36: Use a graphing calculator to find the value of each determinant.
 4.5.37: Write a matrix whose determinant is zero.
 4.5.38: Khalid and Erica are finding the determinant of 8 5 3 2 . Who is c...
 4.5.39: Find a counterexample to disprove the following statement. Two diff...
 4.5.40: Find a thirdorder determinant in which no element is 0, but for wh...
 4.5.41: Use the information about the Bermuda Triangle on page 194 to expla...
 4.5.42: Find the area of triangle ABC. A 10 units 2 B 12 un its 2 C 14 u ni...
 4.5.43: Use the table to determine the expression that best represents the ...
 4.5.44: For Exercises 44 and 45, use the following information. (Lesson 44...
 4.5.45: For Exercises 44 and 45, use the following information. (Lesson 44...
 4.5.46: Find each product, if possible
 4.5.47: Find each product, if possible
 4.5.48: Find each product, if possible
 4.5.49: The length of a marathon was determined in the 1908 Olympic Games i...
 4.5.50: Write an equation in slopeintercept form for the line that satisfi...
 4.5.51: Write an equation in slopeintercept form for the line that satisfi...
 4.5.52: Write an equation in slopeintercept form for the line that satisfi...
 4.5.53: Write an equation in slopeintercept form for the line that satisfi...
 4.5.54: Solve each system of equations
 4.5.55: Solve each system of equations
 4.5.56: Solve each system of equations
Solutions for Chapter 4.5: Determinants
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 4.5: Determinants
Get Full SolutionsSince 56 problems in chapter 4.5: Determinants have been answered, more than 47718 students have viewed full stepbystep solutions from this chapter. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. Chapter 4.5: Determinants includes 56 full stepbystep solutions.

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

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

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.

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

Solvable system Ax = b.
The right side b is in the column space of A.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·