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Matrices

Multiply matrices

Multiply matrix a by matrix b matrix a: row 1 = [0 1 2] row 2 = [-1 4 .5] row 3 = [1 3 0] matrix b: row 1 = [3 -1 5] row 2 = [0 2 2] row 3 = [4 -6 0]

Matrix operations

Let AX =C, If B is the inverse of A, then the solve the matrix X Matix B(3x3)row 1 = -17 78 24 row 2 = -3 13 4 row 3 = -10 16 5 Matrix C(1x3) row 1 = 24 row 2 = -686 row 3 = 2246

Matrix operations

#5 M is a (3*3) matrix, use the identity matrix to find the inverse of M= 1 5 42 -7 -34 -287 28 136 1149

Matrix Series

Please see the attached file for the fully formatted problems. Let K be an nxn matrix and .. a small number. Imitating .... valid for small x, it is natural to define .... Explain why this makes sense. Prove trace log... = log det.... Still with ... small so that everything makes sense.

Differential Operator

Attached deals with the orthogonality of the eigenfunctions of the self adjoint operator.

Diagonal Matrix

Diagoalize te given matrix: A= 3 1 1 1 0 2 1 2 0 an find an orthogonal matrix P such that P'-1(Pinverse)AP is diagonal.

Orthogonal Matrix

Verify that P= 2/3 -2/3 1/3 2/3 1/3 -2/3 1/3 2/3 2/3 is orthogonal matrix.

Symmetric matrix

What is the symmetric matrix of the quadratic form 2x^2 - 8xy + 4y^2?

Determinants

Find the value of each of the third-order determinants:  2 0 -2   1 1 5  (this is a  3 4 5  3x3 matrix)

Matrix Proof: Determinants

Please see the attached file for the full problem description: 1. Show that if   0, then A^-1=1/ [ d -b ] [ -c a ] . Help: : is the determinant A [ d -b] [ -c a ] : is a 2 x 2 matrices

Matrix subspace

Consider the matrix a=(1 1 2 1 2 -1 3 2 1 5 5 2) Find N(A), R(A), N(A^T),R(A^T). Show that the fundamental subspace theorem holds: N(A^T)=R(A)^(upside down T), N(A)=R(A^T)^(upsidedown T). Hint: Notice that the fourth row is the sum of the first three rows.

Solution for an IVP differential equation problem

Using the method of undetermined coefficients, find the solution of the system: X'=AX + B that satisfies the initial condition: X(0)=( 0 1 -1). A and B are matrices defined in the attached Notepad file. Note: When solving the homogeneous soln, exhibit a fundamental matrix psi(t) and al