The EPGY Theorem Proving Environment

Student Exercises

Linear Algebra (M51a)

Matrix algebra exercises

• Matrix addition is commutative.
• The transpose of the sum of two matrices is the sum of the transposes.
• The transpose of the product of two matrices is the product of the transposes, multiplied in the opposite order.
• The trace of the matrix AB equals the trace of the matrix BA
• If A is an invertible matrix, then the trace of the matrix B equals the trace of the matrix ABA^-1.

More matrix exercises

• If AB = I_n and BA = I_m, then B = A^-1. (Matrix inverses are unique.)
• If S and T are (n×n)-matrices and both are invertible, then ST is invertible and (ST)^-1 = T^-1S^-1.
• If B is a square matrix, then B + B^T is symmetric.
• If B is a square matrix, then B - B^T is skew-symmetric.
• If B is a square matrix, then BB^T is symmetric.
• Any square matrix can be uniquely decomposed into the sum of a symmetric and a skew-symmetric matrix. This exercise is divided into two parts:
  1. If A is a square matrix, then there are a symmetric matrix U and a skew-symmetric matrix V such that A = U + V. (Existence)
  2. If A is a square matrix, U is a symmetric matrix, V is a skew-symmetric matrix, and A = U + V, then U = ½(A + A^T) and V = ½(A - A^T). (Uniqueness)

Linear independence exercises

Eigenvector/eigenvalue exercises