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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.