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Subsection 1.3 Algebraic properties of matrix operations

Subsubsection 1.3.1 Properties of matrix sum

For the properties shown below, \(\boldsymbol{A}_{m\times n}, \boldsymbol{B}_{m\times n}, \text{ and } \boldsymbol{C}_{m\times n} \) are matrices and \(r \text{ and } s \) are scalars.
  • Commutative property.
    The sum of two matrices is commutative:
    \begin{equation*} \boldsymbol{A}_{m\times n} + \boldsymbol{B}_{m\times n} = \boldsymbol{B}_{m\times n} + \boldsymbol{A}_{m\times n} \end{equation*}
    \begin{equation*} \boldsymbol{A} = \left[ \begin{array}{ccc} 4 \amp 3 \amp 1\\ -1 \amp 0 \amp -2\\ \end{array} \right], \hspace{1cm} \boldsymbol{B} = \left[ \begin{array}{ccc} 2 \amp 0 \amp -1\\ 3 \amp 5 \amp -1\\ \end{array} \right]. \end{equation*}
    \begin{equation*} \boldsymbol{A}+ \boldsymbol{B} = \left[ \begin{array}{ccc} 4 \amp 3 \amp 1\\ -1 \amp 0 \amp -2\\ \end{array} \right] \hspace{0.2cm} + \hspace{0.2cm} \left[ \begin{array}{ccc} 2 \amp 0 \amp -1\\ 3 \amp 5 \amp -1\\ \end{array} \right] \hspace{0.5cm} = \hspace{0.5cm} \left[ \begin{array}{ccc} 6 \amp 3 \amp 0\\ 2 \amp 5 \amp -3\\ \end{array} \right]. \end{equation*}
    \begin{equation*} \boldsymbol{B}+ \boldsymbol{A} = \left[ \begin{array}{ccc} 2 \amp 0 \amp -1\\ 3 \amp 5 \amp -1\\ \end{array} \right] \hspace{0.2cm} + \hspace{0.2cm} \left[ \begin{array}{ccc} 4 \amp 3 \amp 1\\ -1 \amp 0 \amp -2\\ \end{array} \right] \hspace{0.5cm} = \hspace{0.5cm} \left[ \begin{array}{ccc} 6 \amp 3 \amp 0\\ 2 \amp 5 \amp -3\\ \end{array} \right]. \end{equation*}
  • Associative property.
    Matrix addition is associative :
    \begin{equation*} \boldsymbol{A}_{m\times n} + \left( \boldsymbol{B}_{m\times n} + \boldsymbol{C}_{m\times n} \right) = \left(\boldsymbol{A}_{m\times n} + \boldsymbol{B}_{m\times n}\right) + \boldsymbol{C}_{m\times n}. \end{equation*}
    If we have more than two matrices to add, we can first sum two of them and then add the third matrix to this result. This property tells us that the order in which we choose the first two matrices does not matter.
    Consider the following three matrices
    \begin{equation*} \boldsymbol{A} = \left[ \begin{array}{cc} 2 \amp 1 \\ 0 \amp 3\\ \end{array}\right], \hspace{1cm} \boldsymbol{B} = \left[ \begin{array}{cc} -1 \amp 0 \\ 1 \amp 2\\ \end{array}\right], \hspace{1cm} \boldsymbol{C} = \left[ \begin{array}{cc} 0 \amp -4 \\ 1 \amp -2\\ \end{array}\right]. \end{equation*}
    • Note that \(\boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} \) can be obtained by first adding \(\boldsymbol{A} \) and \(\boldsymbol{B}\) and then adding the resulting matrix to \(\boldsymbol{C}: \)
      \begin{equation*} \boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} = \left(\boldsymbol{A} + \boldsymbol{B}\right) + \boldsymbol{C}. \end{equation*}
      \begin{equation*} \left(\boldsymbol{A} + \boldsymbol{B}\right) = \left[ \begin{array}{cc} 2 \amp 1 \\ 0 \amp 3\\ \end{array}\right] + \left[ \begin{array}{cc} -1 \amp 0 \\ 1 \amp 2\\ \end{array}\right] = \left[ \begin{array}{cc} 1 \amp 1 \\ 1 \amp 5\\ \end{array}\right] \end{equation*}
      \begin{equation*} \left(\boldsymbol{A} + \boldsymbol{B}\right) + \boldsymbol{C} = \left[ \begin{array}{cc} 1 \amp 1 \\ 1 \amp 5\\ \end{array}\right] + \left[ \begin{array}{cc} 0 \amp -4 \\ 1 \amp -2\\ \end{array}\right]= \left[ \begin{array}{cc} 1 \amp -3 \\ 2 \amp 3\\ \end{array}\right]. \end{equation*}
    • Alternatively, \(\boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} \) can be obtained by first adding \(\boldsymbol{B} \) and \(\boldsymbol{C}\) and then adding the resulting matrix to \(\boldsymbol{A}: \)
      \begin{equation*} \boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} = \boldsymbol{A} + \left(\boldsymbol{B} + \boldsymbol{C}\right). \end{equation*}
      \begin{equation*} \left(\boldsymbol{B} + \boldsymbol{C}\right) = \left[ \begin{array}{cc} -1 \amp 0 \\ 1 \amp 2\\ \end{array}\right] + \left[ \begin{array}{cc} 0 \amp -4 \\ 1 \amp -2\\ \end{array}\right] = \left[ \begin{array}{cc} -1 \amp -4 \\ 2 \amp 0\\ \end{array}\right] \end{equation*}
      \begin{equation*} \boldsymbol{A} + \left(\boldsymbol{B}+ \boldsymbol{C} \right) = \left[ \begin{array}{cc} 2 \amp 1 \\ 0 \amp 3\\ \end{array}\right] + \left[ \begin{array}{cc} -1 \amp -4 \\ 2 \amp 0\\ \end{array}\right]= \left[ \begin{array}{cc} 1 \amp -3 \\ 2 \amp 3\\ \end{array}\right]. \end{equation*}
    • Another alternative is to find \(\boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} \) by first adding \(\boldsymbol{A} \) and \(\boldsymbol{C}\) and then adding the resulting matrix to \(\boldsymbol{B}: \)
      \begin{equation*} \boldsymbol{A} + \boldsymbol{B} + \boldsymbol{C} = \left( \boldsymbol{A} + \boldsymbol{C}\right) + \boldsymbol{B}. \end{equation*}
      \begin{equation*} \left(\boldsymbol{A} + \boldsymbol{C}\right) = \left[ \begin{array}{cc} 2 \amp 1 \\ 0 \amp 3\\ \end{array}\right] + \left[ \begin{array}{cc} 0 \amp -4 \\ 1 \amp -2\\ \end{array}\right] = \left[ \begin{array}{cc} 2 \amp -3 \\ 1 \amp 1\\ \end{array}\right] \end{equation*}
      \begin{equation*} \left(\boldsymbol{A} + \boldsymbol{C} \right)+ \boldsymbol{B} = \left[ \begin{array}{cc} 2 \amp -3 \\ 1 \amp 1\\ \end{array}\right] + \left[ \begin{array}{cc} -1 \amp 0 \\ 1 \amp 2\\ \end{array}\right]= \left[ \begin{array}{cc} 1 \amp -3 \\ 2 \amp 3\\ \end{array}\right]. \end{equation*}

Subsubsection 1.3.2 Properties of matrix-scalar multiplication

For the properties shown below, \(\boldsymbol{A}_{m\times n}, \boldsymbol{B}_{m\times n}, \text{ and } \boldsymbol{C}_{m\times n} \) are matrices and \(r \text{ and } s \) are scalars.
  • Distributive property 1.
    Multiplication of a matrix by a scalar is distributive with respect to matrix addition:
    \begin{equation*} r \left( \boldsymbol{A}_{m\times n} + \boldsymbol{B}_{m\times n}\right) = r\, \boldsymbol{A}_{m\times n} + r\, \boldsymbol{B}_{m\times n}. \end{equation*}
    \begin{equation*} \begin{array}{ccc} 4 \times \left( \left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 4 \\ 1 \amp 3 \\ -2 \amp 1 \\ \end{array}\right] + \left[ \begin{array}{rr} 3 \amp -1 \\ 1 \amp 1 \\ 2 \amp -4 \\ -2 \amp 2 \\ \end{array}\right] \right) \amp = \amp 4 \times \left[ \begin{array}{rr} -1 \amp 2 \\ 0 \amp 4 \\ 1 \amp 3 \\ -2 \amp 1 \\ \end{array}\right] + 4 \times \left[ \begin{array}{rr} 3 \amp -1 \\ 1 \amp 1 \\ 2 \amp -4 \\ -2 \amp 2 \\ \end{array}\right] \\ \end{array} \end{equation*}
    \begin{equation*} \hspace{1.5cm} \begin{array}{ccc} 4 \times \left( \left[ \begin{array}{rr} 2 \amp 1 \\ 1 \amp 5 \\ 3 \amp -1 \\ -4 \amp 3 \\ \end{array}\right] \right) \amp \hspace{0.9cm} = \hspace{0.9cm} \amp \left[ \begin{array}{rr} -4 \amp 8 \\ 0 \amp 16 \\ 4 \amp 12 \\ -8 \amp 4 \\ \end{array}\right] + \left[ \begin{array}{rr} 12 \amp -4 \\ 4 \amp 4 \\ 8 \amp -16 \\ -8 \amp 8 \\ \end{array}\right] \\ \amp \amp\\ \left[ \begin{array}{rr} 8 \amp 4 \\ 4 \amp 20 \\ 12 \amp -4 \\ -16 \amp 12 \\ \end{array}\right] \amp \hspace{0.9cm} = \hspace{0.9cm} \amp \left[ \begin{array}{rr} 8 \amp 4 \\ 4 \amp 20 \\ 12 \amp -4 \\ -16 \amp 12 \\ \end{array}\right] \\ \end{array} \end{equation*}
  • Distributive property 2.
    Multiplication of a matrix by a scalar is distributive with respect to the scalar addition:
    \begin{equation*} \left(r+s \right) \boldsymbol{C}_{m\times n} = r\, \boldsymbol{C}_{m\times n} + s\, \boldsymbol{C}_{m\times n}. \end{equation*}
    \begin{equation*} \begin{array}{l} (2 + 4) \times \left[\begin{array}{rr} -2 \amp 1 \\ 0 \amp 1 \\ \end{array}\right] = 2 \times \left[\begin{array}{rr} -2 \amp 1 \\ 0 \amp 1 \\ \end{array}\right] + 4 \times \left[\begin{array}{rr} -2 \amp 1 \\ 0 \amp 1 \\ \end{array}\right]\\ \amp \amp \\ \hspace{0.7cm} 6 \times \left[\begin{array}{rr} -2 \amp 1 \\ 0 \amp 1 \\ \end{array}\right] \hspace{0.5cm} = \hspace{0.5cm} \left[\begin{array}{rr} -4 \amp 2 \\ 0 \amp 2 \\ \end{array}\right] \hspace{0.3cm} + \hspace{0.3cm} \left[\begin{array}{rr} -8 \amp 4 \\ 0 \amp 4 \\ \end{array}\right]\\ \amp \amp \\ \hspace{0.7cm}\left[\begin{array}{rr} -12 \amp 6 \\ 0 \amp 6 \\ \end{array}\right] \hspace{1.1cm} = \hspace{1.5cm} \left[\begin{array}{rr} -12 \amp 6 \\ 0 \amp 6 \\ \end{array}\right] \\ \end{array} \end{equation*}
  • Associative property.
    Multiplication of a matrix by a scalar is associative with respect to the scalar multiplication:
    \begin{equation*} r \left(s \times \boldsymbol{C}_{m\times n} \right) = (r\, s) \boldsymbol{C}_{m\times n}. \end{equation*}
    \begin{equation*} \begin{array}{l} -1 \times \left( 2\times \left[\begin{array}{rrr} 1 \amp -1 \amp 3 \\ 2 \amp 0 \amp -1 \\ 3 \amp 0 \amp 0\\ \end{array}\right] \right) = (-1 \times 2 ) \times \left[\begin{array}{rrr} 1 \amp -1 \amp 3 \\ 2 \amp 0 \amp -1 \\ 3 \amp 0 \amp 0\\ \end{array}\right] \\ \amp \amp \\ \hspace{0.7cm} -1 \times \left[\begin{array}{rrr} 2 \amp -2 \amp 6 \\ 4 \amp 0 \amp -2 \\ 6 \amp 0 \amp 0\\ \end{array}\right] \hspace{0.9cm} = \hspace{0.9cm} -2 \times \left[\begin{array}{rrr} 1 \amp -1 \amp 3 \\ 2 \amp 0 \amp -1 \\ 3 \amp 0 \amp 0\\ \end{array}\right]\\ \amp \amp \\ \hspace{1.1cm}\left[\begin{array}{rrr} -2 \amp 2 \amp -6 \\ -4 \amp 0 \amp 2 \\ -6 \amp 0 \amp 0\\ \end{array}\right] \hspace{1.6cm} = \hspace{1cm} \left[\begin{array}{rrr} -2 \amp 2 \amp -6 \\ -4 \amp 0 \amp 2 \\ -6 \amp 0 \amp 0\\ \end{array}\right] \\ \end{array} \end{equation*}

Subsubsection 1.3.3 Properties of matrix-matrix multiplication

For the properties shown below, \(\boldsymbol{A}, \boldsymbol{B}, \text{ and } \boldsymbol{C} \) are matrices and \(r \) is a scalar. The sizes for the matrices will be given for each property.
  • Associative property.
    Matrix-matrix multiplication is associative:
    \begin{equation*} \begin{array}{ccc} \boldsymbol{A} \left(\boldsymbol{B}\,\boldsymbol{C}\right) \amp = \amp \left(\boldsymbol{A}\,\boldsymbol{B}\right)\,\boldsymbol{C}\\ \amp \amp \\ \boldsymbol{A}_{m\times n} \left(\boldsymbol{B}_{n\times p} \boldsymbol{C}_{p\times q}\right) \amp = \amp \left(\boldsymbol{A}_{m\times n} \boldsymbol{B}_{n\times p}\right) \boldsymbol{C}_{p\times q}.\\ \end{array} \end{equation*}
    \begin{equation*} \begin{array}{ccc} \boldsymbol{A}_{3\times 2} \left(\boldsymbol{B}_{2\times 4} \boldsymbol{C}_{4\times 3}\right) \amp = \amp \left(\boldsymbol{A}_{3\times 2} \boldsymbol{B}_{2\times 4}\right) \boldsymbol{C}_{4\times 3}\\ \amp \amp \\ \left[ \begin{array}{rr} 1 \amp 1\\ -2 \amp 0 \\ 3 \amp 1\\ \end{array}\right]\, \left( \left[ \begin{array}{rrrr} 0 \amp 1 \amp -1 \amp 3\\ 2 \amp -1 \amp 1 \amp -2\\ \end{array}\right]\, \left[ \begin{array}{rrr} 1 \amp -1 \amp -3\\ 2 \amp 3 \amp 1 \\ 4 \amp 1 \amp 0 \\ 0 \amp -2 \amp 1 \\ \end{array}\right] \right) \amp = \amp \left( \left[ \begin{array}{rr} 1 \amp 1\\ -2 \amp 0 \\ 3 \amp 1\\ \end{array}\right] \, \left[ \begin{array}{rrrr} 0 \amp 1 \amp -1 \amp 3\\ 2 \amp -1 \amp 1 \amp -2\\ \end{array}\right] \right)\, \left[ \begin{array}{rrr} 1 \amp -1 \amp -3\\ 2 \amp 3 \amp 1 \\ 4 \amp 1 \amp 0 \\ 0 \amp -2 \amp 1 \\ \end{array}\right] \end{array} \end{equation*}
    \begin{equation*} \begin{array}{lll} \hspace{2cm} \left[ \begin{array}{rr} 1 \amp 1\\ -2 \amp 0 \\ 3 \amp 1\\ \end{array}\right]\, \left[ \begin{array}{rrr} -6 \amp -5 \amp 4 \\ 4 \amp 0 \amp -9 \\ \end{array}\right] \hspace{2.8cm} \amp = \amp \hspace{1.7cm} \left[ \begin{array}{rrrr} 2 \amp 0 \amp -1 \amp 1 \\ 0 \amp -2 \amp 4 \amp -6 \\ 2 \amp 2 \amp -5 \amp 7 \\ \end{array}\right]\, \left[ \begin{array}{rrr} 1 \amp -1 \amp -3\\ 2 \amp 3 \amp 1 \\ 4 \amp 1 \amp 0 \\ 0 \amp -2 \amp 1 \\ \end{array}\right]\\ \amp \amp \\ \hspace{2.2cm} \left[ \begin{array}{rrr} -2 \amp -5 \amp -5 \\ 12 \amp 10 \amp -8 \\ -14 \amp -15 \amp 3 \\ \end{array}\right]_{3 \times 3} \amp = \amp \hspace{2.2cm} \left[ \begin{array}{rrr} -2 \amp -5 \amp -5 \\ 12 \amp 10 \amp -8 \\ -14 \amp -15 \amp 3 \\ \end{array}\right]_{3 \times 3} \end{array} \end{equation*}
  • Distributive property.
    Matrix-matrix multiplication is distributive with respect to matrix addition:
    • Case 1:
      \begin{equation*} \begin{array}{ccc} \boldsymbol{A}\,\left(\boldsymbol{B} + \boldsymbol{C} \right) \amp = \amp \boldsymbol{A}\,\boldsymbol{B} + \boldsymbol{A}\,\boldsymbol{C} \amp \amp\\ \boldsymbol{A}_{m\times n}\,\left(\boldsymbol{B}_{n\times p} + \boldsymbol{C}_{n\times p} \right) \amp = \amp \boldsymbol{A}_{m\times n}\,\boldsymbol{B}_{n\times p} + \boldsymbol{A}_{m\times n}\,\boldsymbol{C}_{n\times p} \end{array} \end{equation*}
      \begin{equation*} \begin{array}{ccc} \boldsymbol{A}_{3\times 2}\,\left(\boldsymbol{B}_{2\times 4} + \boldsymbol{C}_{2\times 4}\right) \amp = \amp \boldsymbol{A}_{3\times 2}\,\boldsymbol{B}_{2\times 4} + \boldsymbol{A}_{3\times 2}\,\boldsymbol{C}_{2\times 4}\\ \amp \amp \\ \left[ \begin{array}{rr} 1 \amp 1\\ -2 \amp 0 \\ 3 \amp 1\\ \end{array}\right]\, \left( \left[ \begin{array}{rrrr} 0 \amp 1 \amp -1 \amp 3\\ 2 \amp -1 \amp 1 \amp -2\\ \end{array}\right] + \left[ \begin{array}{rrrr} 1 \amp -1 \amp -3 \amp 1\\ 2 \amp 3 \amp 1 \amp 0 \\ \end{array}\right] \right) \amp = \amp \left[ \begin{array}{rr} 1 \amp 1\\ -2 \amp 0 \\ 3 \amp 1\\ \end{array}\right]\, \left[ \begin{array}{rrrr} 0 \amp 1 \amp -1 \amp 3\\ 2 \amp -1 \amp 1 \amp -2\\ \end{array}\right] + \left[ \begin{array}{rr} 1 \amp 1\\ -2 \amp 0 \\ 3 \amp 1\\ \end{array}\right]\, \left[ \begin{array}{rrrr} 1 \amp -1 \amp -3 \amp 1\\ 2 \amp 3 \amp 1 \amp 0 \\ \end{array}\right] \end{array} \end{equation*}
      \begin{equation*} \begin{array}{lll} \left[ \begin{array}{rr} 1 \amp 1\\ -2 \amp 0 \\ 3 \amp 1\\ \end{array}\right]\, \hspace{1cm} \left[ \begin{array}{rrrr} 1 \amp 0 \amp-4 \amp 4\\ 4 \amp 2 \amp 2 \amp -2\\ \end{array}\right] \hspace{4.3cm} \amp = \amp \hspace{1cm}\left[ \begin{array}{rrrr} 2 \amp 0 \amp 0 \amp 1\\ 0 \amp -2 \amp 2 \amp -6\\ 2 \amp 2 \amp -2 \amp 7\\ \end{array}\right] \hspace{1cm} + \hspace{0.4cm} \left[ \begin{array}{rrrr} 3 \amp 2 \amp -2 \amp 1\\ -2 \amp 2 \amp 6 \amp -2\\ 5 \amp 0 \amp -8 \amp 3\\ \end{array}\right]\\ \amp \amp\\ \hspace{5cm}\left[ \begin{array}{rrrr} 5 \amp 2 \amp -2 \amp 2\\ -2 \amp 0 \amp 8 \amp -8\\ 7 \amp 2 \amp -10 \amp 10\\ \end{array}\right] \amp = \amp \hspace{2cm}\left[ \begin{array}{rrrr} 5 \amp 2 \amp -2 \amp 2\\ -2 \amp 0 \amp 8 \amp -8\\ 7 \amp 2 \amp -10 \amp 10\\ \end{array}\right] \end{array} \end{equation*}
    • Case 2:
      \begin{equation*} \begin{array}{ccc} \left( \boldsymbol{A} + \boldsymbol{B}\right)\,\boldsymbol{C} \amp = \amp \boldsymbol{A}\,\boldsymbol{C} + \boldsymbol{B}\,\boldsymbol{C} \amp \amp\\ \left( \boldsymbol{A}_{m\times n} + \boldsymbol{B}_{m\times n}\right)\,\boldsymbol{C}_{n\times p} \amp = \amp \boldsymbol{A}_{m\times n}\,\boldsymbol{C}_{n\times p} + \boldsymbol{B}_{m\times n}\,\boldsymbol{C}_{n\times p} \end{array} \end{equation*}
      \begin{equation*} \begin{array}{ccc} \left( \boldsymbol{A}_{2\times 2} + \boldsymbol{B}_{2\times 2}\right)\,\boldsymbol{C}_{2\times 3} \amp = \amp \boldsymbol{A}_{2\times 2}\,\boldsymbol{C}_{2\times 3} + \boldsymbol{B}_{2\times 2}\,\boldsymbol{C}_{2\times 3}\\ \amp \amp \\ \left( \left[ \begin{array}{rr} 1 \amp 0\\ 0 \amp 1 \\ \end{array}\right] + \left[ \begin{array}{rr} 3 \amp -2\\ 1 \amp 4 \\ \end{array}\right] \right)\, \left[ \begin{array}{rrr} 2 \amp 1 \amp 0\\ 1 \amp 3 \amp -1 \end{array}\right] \amp = \amp \left[ \begin{array}{rr} 1 \amp 0\\ 0 \amp 1 \\ \end{array}\right]\, \left[ \begin{array}{rrr} 2 \amp 1 \amp 0\\ 1 \amp 3 \amp -1 \end{array}\right] + \left[ \begin{array}{rr} 3 \amp -2\\ 1 \amp 4 \\ \end{array}\right]\, \left[ \begin{array}{rrr} 2 \amp 1 \amp 0\\ 1 \amp 3 \amp -1 \end{array}\right] \end{array} \end{equation*}
      \begin{equation*} \begin{array}{lll} \hspace{1cm}\left[ \begin{array}{rr} 4 \amp -2\\ 1 \amp 5 \\ \end{array}\right]\, \left[ \begin{array}{rrr} 2 \amp 1 \amp 0\\ 1 \amp 3 \amp -1 \end{array}\right]\hspace{1.5cm} \amp = \amp \hspace{1cm} \left[ \begin{array}{rrr} 2 \amp 1 \amp 0\\ 1 \amp 3 \amp -1 \end{array}\right] + \left[ \begin{array}{rrr} 4 \amp -3 \amp 2\\ 6 \amp 13 \amp -4 \end{array}\right]\\ \amp \amp\\ \hspace{2cm}\left[ \begin{array}{rrr} 6 \amp -2 \amp 2\\ 7 \amp 16 \amp -5 \end{array}\right] \amp = \amp \hspace{2cm}\left[ \begin{array}{rrr} 6 \amp -2 \amp 2\\ 7 \amp 16 \amp -5 \end{array}\right] \end{array} \end{equation*}
  • Associative property with scalar multiplication.
    Matrix multiplication is associative with respect to scalar multiplication:
    \begin{equation*} \begin{array}{cll} r \left(\boldsymbol{A}\,\boldsymbol{B}\right) \amp = \left(r\,\boldsymbol{A}\right)\,\boldsymbol{B} \amp = \boldsymbol{A}\,\left(r\,\boldsymbol{B} \right)\\ \amp \amp\\ r \left(\boldsymbol{A}_{m\times n}\,\boldsymbol{B}_{n\times p}\right) \amp = \left(r\,\boldsymbol{A}\right)_{m\times n}\,\boldsymbol{B}_{n\times p} \amp = \boldsymbol{A}_{m\times n}\,\left(r\,\boldsymbol{B} \right)_{n\times p} \end{array} \end{equation*}
    \begin{equation*} \begin{array}{cll} 2 \left(\boldsymbol{A}_{2\times 2}\,\boldsymbol{B}_{2\times 2}\right) \amp = \left(2\,\boldsymbol{A}\right)_{2\times 2}\,\boldsymbol{B}_{2\times 2} \amp = \boldsymbol{A}_{2\times 2}\,\left(2\,\boldsymbol{B} \right)_{2\times 2} \amp \amp \\ \amp \amp \\ 2 \left( \left[ \begin{array}{rr} 3 \amp 2\\ 1 \amp 0\\ \end{array}\right]\, \left[ \begin{array}{rr} 1 \amp -1\\ 2 \amp 1\\ \end{array}\right] \right) \amp = \left( 2\,\left[ \begin{array}{rr} 3 \amp 2\\ 1 \amp 0\\ \end{array}\right] \right)\, \left[ \begin{array}{rr} 1 \amp -1\\ 2 \amp 1\\ \end{array}\right] \amp = \left[ \begin{array}{rr} 3 \amp 2\\ 1 \amp 0\\ \end{array}\right]\, \left( 2\,\left[ \begin{array}{rr} 1 \amp -1\\ 2 \amp 1\\ \end{array}\right] \right)\\ \amp \amp \\ 2\, \left[ \begin{array}{rr} 7 \amp -1\\ 1 \amp -1\\ \end{array}\right] \amp = \left[ \begin{array}{rr} 6 \amp 4\\ 3 \amp 0\\ \end{array}\right]\, \left[ \begin{array}{rr} 1 \amp -1\\ 2 \amp 1\\ \end{array}\right] \amp = \left[ \begin{array}{rr} 3 \amp 2\\ 1 \amp 0\\ \end{array}\right]\, \left[ \begin{array}{rr} 2 \amp -2\\ 4 \amp 2\\ \end{array}\right]\\ \end{array} \end{equation*}
    \begin{equation*} \begin{array}{lll} \amp \amp \\ \left[ \begin{array}{rr} 14 \amp -2\\ 2 \amp -2\\ \end{array}\right] \hspace{0.5cm} \amp = \hspace{1cm}\left[ \begin{array}{rr} 14 \amp -2\\ 2 \amp -2\\ \end{array}\right]\hspace{1cm} \amp = \hspace{1cm}\left[ \begin{array}{rr} 14 \amp -2\\ 2 \amp -2\\ \end{array}\right] \end{array} \end{equation*}
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