14  Matrix Algebra Review

Appendix: Optional Review of Matrix Facts

Matrix dimensions

If \(\mathbf{A}\) is \(m \times n\) and \(\mathbf{B}\) is \(n \times p\), then \(\mathbf{A}\mathbf{B}\) is defined and is \(m \times p\).

Transpose rules

For conformable matrices,

\[ (\mathbf{A}\mathbf{B})^\top = \mathbf{B}^\top \mathbf{A}^\top. \]

Inverse rule

If \(\mathbf{A}\) and \(\mathbf{B}\) are invertible, then

\[ (\mathbf{A}\mathbf{B})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}. \]

Symmetric matrices

A matrix \(\mathbf{A}\) is symmetric if

\[ \mathbf{A}^\top = \mathbf{A}. \]

Covariance matrices are always symmetric.