Solve the pair of differential equations

	x'1 (t) = x1 (t) + x2 (t)
	x'2 (t) = - x1 (t) + x2 (t)

with initial conditions x₁(0) = x₂(0) = 1000.

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Solve the initial value problem

by converting it into a system of differential equations of order 1 and then solving the system.

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Use Gershgorin discs to study the location of the eigenvalues of the matrix

	2-j	2	1
	1	2j	0
	j	1	2

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Show that if is an eigenvalue of the matrix

	2	-1	0	0
	-1	2	-1	0
	0	-1	2	-1
	0	0	-1	2

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Consider the matrix

A =		2	-2	1
		-1	3	-1
		-2	-4	3

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Compute the norms ||A||₁ ,  ||A|| and ||A||_Fr of the matrix

A =		-1	1	0	-1
		0	1	-2	1
		1	0	1	-1
		-2	0	-1	1

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Consider the matrix A of the previous exercise. Compute ||A||₂ iteratively. Start from the vector ₀ = (0, 1, - 1, 1).

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Compute the condition number of A with respect to 1-norm and -norm. Here

A =		0.501	-0.343		.
		-0.872	0.597

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Compute e^A, where

a) A = 3 -2 , -1 4

b) A = 0 1 0 0 . 0 0 1 0 0 0 0 1 0 0 0 0

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Compute the spectral radius of the matrix

A =		1	0	0
		0	1	-1
		-1	-1	2