Applied Linear Algebra

Subsubsection 1.1.1.1 Matrix size

The size of a matrix is given by the number of its rows and its columns. A \(m \times n \) matrix has \(m\) rows and \(n \) columns. In general, we place the size as a subscript of the name of the matrix, \(\boldsymbol{A}_{m\times n}.\)

Subsubsection 1.1.1.2 Matrix elements

We identify the elements in a matrix based on their location with respect to the rows and columns of the matrix,

Subsubsection 1.1.1.3 Matrix equality

Two matrices are equal when they have the same numbers in corresponding entries.

Subsubsection 1.1.1.4 Square matrix

A square matrix has the same number of rows and columns, \(m=n. \) The elements of a square matrix \(\boldsymbol{A}_{n\times n} \) where the subscripts are equal \(\left( a_{11}, a_{22}, \cdots, a_{nn}\right) \) are called the matrix main diagonal.

Subsubsection 1.1.1.5 Identity matrix

An identity matrix is a square matrix with ones in its main diagonal and zeros everywhere else. We use the notation \(\boldsymbol{I}_n \) to indicate an identity matrix that is \(n \times n. \)

Subsubsection 1.1.1.6 Zero matrix

A zero matrix is a matrix filled with all zeros and denoted by \(\boldsymbol{0}_{m\times n}. \)

Subsubsection 1.1.2.1 Matrix addition

Let \(\boldsymbol{A}_{m\times n} \) and \(\boldsymbol{B}_{m\times n} \) be two matrices of the same size. The sum \(\boldsymbol{C}_{m\times n} = \boldsymbol{A} + \boldsymbol{B} \) is obtained by adding the corresponding elements of \(\boldsymbol{A} \) and \(\boldsymbol{B}: \) \(c_{ij} = a_{ij} + b_{ij}. \) If \(\boldsymbol{A} \) and \(\boldsymbol{B} \) are not the same size the sum does not exists.

Subsubsection 1.1.2.2 Matrix-scalar multiplication

Let \(\boldsymbol{A}_{m \times n} \) be a matrix and \(c\) a scalar. The scalar multiple of \(\boldsymbol{A}\) by \(c\text{,}\) denoted \(c\boldsymbol{A}\text{,}\) is the \(m \times n\) matrix obtained by multiplying every element of \(\boldsymbol{A}\) by \(c\text{,}\) \(\boldsymbol{B} = c\boldsymbol{A} \Rightarrow b_{ij} = c \cdot a_{ij}.\)

Subsubsection 1.1.2.3 Matrix-matrix multiplication

The matrix product of the matrix \(\boldsymbol{A}_{m \times n} \) and the matrix \(\boldsymbol{B}_{n \times r}, \) is the matrix \(\boldsymbol{C}_{m \times r} = \boldsymbol{A}_{m \times n} \times \boldsymbol{B}_{n \times r},\) where

That is, the \(ij \) element of the product results from multiplying the \(i^{th}\) row of the first matrix with the \(j^{th}\) column of the second matrix and adding the entries.

Insight 1.1.13. Size matters.

In general, not every matrix can be multiplied to any other matrix. Consider the example depicted in the figure below,

In order to carry out this operation, we need to multiply the three elements of the row in the first matrix with the three elements of the column in the second matrix.

If, for instance, the second matrix had 2 rows instead of 3, the operation will be incomplete.

For this reason, we can only multiply matrices that have the "right dimensions." The number of columns in the first matrix has to be the same as the number of rows in the second matrix.

Subsubsection 1.1.3.1 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*}

  Example 1.1.15. Commutative property.

  \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*}

  Example 1.1.16. Associative property.

  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 tell 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 1.1.3.2 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*}

  Example 1.1.17. Distributive property 1.

  \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*}

  Example 1.1.18. Distributive property 2.

  \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*}

  Example 1.1.19. Associative property.

  \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 1.1.3.3 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*}

  Example 1.1.20. Associative property.

  \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*}

    Example 1.1.21. Distributive property - Case 1.

    \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*}

    Example 1.1.22. Distributive property - Case 2.

    \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*}

  Example 1.1.23. Associative property with scalar multiplication.

  \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*}

Subsubsection 1.1.5.1 Product of two powers

The following holds for any square matrix \(\boldsymbol{A}_{n \times n} \) and integers \(p \) and \(q\text{,}\)

Subsubsection 1.1.5.2 Power of a power

The following holds for any square matrix \(\boldsymbol{A}_{n \times n} \) and integers \(p \) and \(q\text{,}\)