- 3.6.1: Give the equation that each picture models.
- 3.6.2: Copy and fill in the table to solve the equation as in Example A.
- 3.6.3: Give the next stages of the equation, matching the action taken, to...
- 3.6.4: Complete the tables to solve the equations.
- 3.6.5: Give the additive inverse of each number. a. b. 17 c. 23 d. x
- 3.6.6: A multiplicative inverse is a number or expression that you can mul...
- 3.6.7: Solve these equations. Tell what action you take at each stage. a. ...
- 3.6.8: Mini-Investigation A solution to the equation 10 + 3x = 5 is shown ...
- 3.6.9: Solve the equation 4 + 1.2x = 12.4 by using each method. a. balanci...
- 3.6.10: Solve each equation symbolically using the balancing method. a. 3 +...
- 3.6.11: You can solve familiar formulas for a specific variable. For exampl...
- 3.6.12: An equation can have the variable on both sides. In these cases you...
- 3.6.13: APPLICATION Economy drapes for a certain size window cost $90. They...
- 3.6.14: Run the easy level of the LINES program on your calculator. [ See C...
- 3.6.15: The local bagel store sells a bakers dozen of bagels for $6.49, whi...
Solutions for Chapter 3.6: Solving Equations Using the Balancing Method
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Remove row i and column j; multiply the determinant by (-I)i + j •
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
= Xl (column 1) + ... + xn(column n) = combination of columns.
Outer product uv T
= column times row = rank one matrix.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Pseudoinverse A+ (Moore-Penrose 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).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).