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# Solutions for Chapter 8.7: Cramers Rule

## Full solutions for Advanced Engineering Mathematics | 6th Edition

ISBN: 9781284105902

Solutions for Chapter 8.7: Cramers Rule

Solutions for Chapter 8.7
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##### ISBN: 9781284105902

This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Since 15 problems in chapter 8.7: Cramers Rule have been answered, more than 39792 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.7: Cramers Rule includes 15 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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!

• 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 •

• 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.

• 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.

• 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.

• Network.

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.

• 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.

• 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.

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