Theory of Intelligence
11TH•Mathematics•Matrices And Determinants•4.3 Matrix Properties

4.3 Properties of Matrix Operations

Properties of matrix addition, scalar multiplication, and multiplication.

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4.3 Properties of Matrix Operations

If AA, BB, CC are conformable for the indicated operations and cc, dd are scalars:

  1. A+B=B+AA + B = B + A (commutative addition)
  2. (A+B)+C=A+(B+C)(A + B) + C = A + (B + C) (associative addition)
  3. (cd)A=c(dA)(cd)A = c(dA)
  4. A+O=O+A=AA + O = O + A = A (additive identity)
  5. IA=AI=AIA = AI = A (multiplicative identity)
  6. c(A+B)=cA+cBc(A + B) = cA + cB and (c+d)A=cA+dA(c + d)A = cA + dA
  7. A(BC)=(AB)CA(BC) = (AB)C (associative multiplication)
  8. A(B+C)=AB+ACA(B + C) = AB + AC (left distributive)
  9. (A+B)C=AC+BC(A + B)C = AC + BC (right distributive)
  10. c(AB)=(cA)B=A(cB)c(AB) = (cA)B = A(cB)

Example 3

Find ABAB and BABA if

A=[201142306],B=[1−1023−11−23].A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 4 & 2 \\ 3 & 0 & 6 \end{bmatrix},\quad B = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & -1 \\ 1 & -2 & 3 \end{bmatrix}.

Solution

AB=[3−4311729−1518],BA=[1−4−141229−815]AB = \begin{bmatrix} 3 & -4 & 3 \\ 11 & 7 & 2 \\ 9 & -15 & 18 \end{bmatrix}, \quad BA = \begin{bmatrix} 1 & -4 & -1 \\ 4 & 12 & 2 \\ 9 & -8 & 15 \end{bmatrix}

Thus AB≠BAAB \neq BA. Matrix multiplication is not commutative in general.

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