- 8.7.1: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.2: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.3: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.4: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.5: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.6: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.7: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.8: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.9: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.10: In 1-10, solve the given system of equations by Cramers rule.
- 8.7.11: Use Cramers rule to determine the solution of the system (2 k)x1 kx...
- 8.7.12: Consider the system x1 x2 1 x1 ex2 2. When e is close to 1, the lin...
- 8.7.13: The magnitudes T1 and T2 of the tensions in the support wires shown...
- 8.7.14: The 400-lb block shown in FIGURE 8.7.2 is kept from sliding down th...
- 8.7.15: As shown in FIGURE 8.7.3, a circuit consists of two batteries with ...
Solutions for Chapter 8.7: Cramers Rule
Full solutions for Advanced Engineering Mathematics | 6th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A = CTC = (L.J]))(L.J]))T for positive definite A.
Remove row i and column j; multiply the determinant by (-I)i + j •
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
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.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.