0.1 Practice Problems

1.

Consider matrices \(A\) and \(B\) defined as the following. \[ A = \begin {bmatrix} 1 & 2 \\ 2 & 0 \end {bmatrix}, \quad B = \begin {bmatrix} -1 & 2 \\ 0 & 3 \end {bmatrix} \] Compute the following matrix operations.

(a)

\(A - 2B\)

(b)

\(AB\)

(c)

\((AB - BA)^\intercal \)

2.

Find \(a_2B\) if \(AB\) is defined as the following. \[ AB = \begin {bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end {bmatrix} \]

3.

Find matrix \(A\) such that the following equation holds. \[ 2A - 3A^\intercal = \begin {bmatrix} -1 & -5 \\ 0 & -4 \end {bmatrix} \]

4.

Show that all square matrices can be represented as the sum of symmetric and skew-symmetric matrices.

5.

Show that if \(AB = BA\) for nilpotent matrices \(A\) and \(B\), then the product \(AB\) is nilpotent.

6.

Show that for square matrices \(A\) and \(B\), \(AB + (AB)^\intercal \) is symmetric.

7.

For square matrices \(A\) and \(B\) of order \(n \times n\), show that the following equation holds [MATH02-3]. \[ \det \left ( \begin {bmatrix} A & B \\ B & A \end {bmatrix} \right ) = \det (A + B) \det (A - B) \]

8.

Show that if a square matrix \(A\) is skew-symmetric, then \(A^3\) is also skew-symmetric.