Laws of matrix algebra

In this section, we summarize a few useful properties of the matrix operations we have introduced. Some have been pointed out along the way; some are trivial to check, and some would require a technical proof that we prefer to avoid.

A few properties of matrix addition and multiplication that are formally identical to properties of addition and multiplication of scalars:

• (A + B) + C = A + (B + C)
• A + B = B + A
• (AB)C = A(BC)
• A(B + C) = AB + AC

Matrix multiplication is a bit peculiar since, in general, AB ≠ BA, as we have already pointed out. Other properties involve matrix transposition:

• (A^T)^T = A
• (rA)^T = rA^T
• (A ± B)^T = A^T ± B^T
• (AB)^T = B^T A^T

They are all fairly trivial except for the last one, which we have already pointed out. A few more properties involve inversion. If A and B are square and invertible, as well as of the same dimension, then:

• (A⁻¹)⁻¹ = A
• (A^T)⁻¹ = (A⁻¹)^T
• AB is invertible and (AB)⁻¹ = B⁻¹ A⁻¹