# Linear Algebra Homework 1

## Problem 1 - 1ex

Let

Solve for X the matrix equation

A + 2X = B

Do it first by adding matrices on the left hand side and
then comparing the entries of the

resulting matrix and matrix B. Then solve it again using matrix algebra.

## Problem 2 - 2ex

(a) Let

Compute

(b) Show that for any square matrix is a symmetric matrix.

## Problem 3 - 2ex

Let A be a square matrix. For which combinations of
scalars α and β the matrix

is a symmetric matrix.

## Problem 4 - 1ex

Let A and B are two symmetric matrices. Show that A + B is symmetric.

## Problem 5 - 1ex

Let A and B are two symmetric matrices. For which combinations of scalars α and β the matrix is a symmetric matrix.

## Problem 6 - 1ex

Let A be a symmetric matrix. Show the matrix αI + A is a symmetric matrix.

## Problem 7 - 2ex

Let A and B are two square matrices. Let C = A − B is a
symmetric matrix. What does

that tell you about matrices A and B?

## Problem 8 - 1ex

A matrix is an upper triangular matrix if all its entries
under the diagonal are 0.

Let A and B are two upper triangular matrices. For which combinations of scalars
α and β

the matrix is an upper triangular matrix.