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Subsection 1.1 Matrix basics

In the previous chapter we defined matrices () as arrays of rows and columns, in this section we introduce a convenient notation to work with matrices.

Subsubsection 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, \(\boldsymbol{A}_{m\times n}.\)
\begin{equation*} \boldsymbol{A}_{2\times3} = \left[ \begin{array}{ccc} 3 \amp 0 \amp -3 \\ 2 \amp -4 \amp 5 \\ \end{array} \right] \hspace{2cm} \boldsymbol{B}_{3\times1} = \left[ \begin{array}{c} 3 \\ 2 \\-4 \\ \end{array} \right] \hspace{2cm} \boldsymbol{C}_{2\times2} = \left[ \begin{array}{cc} 1 \amp -2 \\ 0 \amp 3 \\ \end{array} \right] \end{equation*}
Give the size for each matrix.
\begin{equation*} \boldsymbol{A} = \begin{array}{lrrr} \lceil \amp 3 \amp 2 \amp \rceil\\ \vert \amp 5 \amp 0 \amp \vert\\ \lfloor \amp 1 \amp -3 \amp \rfloor\\ \end{array} \end{equation*}
The size of \(\boldsymbol{A}\) is \(\times\)
\begin{equation*} \boldsymbol{B} = \begin{array}{lrrrr} \lceil \amp -1 \amp 4 \amp 0 \amp \rceil \\ \lfloor \amp 2 \amp 1 \amp 9 \amp \rfloor \\ \end{array} \end{equation*}
The size of \(\boldsymbol{B}\) is \(\times\)
\begin{equation*} \boldsymbol{C} = \begin{array}{lrr} \lceil \amp -1 \amp \rceil\\ \vert \amp 2 \amp \vert \\ \vert \amp 4 \amp \vert\\ \lfloor \amp 8 \amp \rfloor \\ \end{array} \end{equation*}
The size of \(\boldsymbol{C}\) is \(\times\)
Answer 1.
\(3\)
Answer 2.
\(2\)
Answer 3.
\(2\)
Answer 4.
\(3\)
Answer 5.
\(4\)
Answer 6.
\(1\)

Subsubsection 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,
For
\begin{equation*} \boldsymbol{A} = \left[ \begin{array}{rrr} 1 \amp 0 \amp -2 \\ 3 \amp -1 \amp 0 \\ 6 \amp 7 \amp 1 \\ \end{array} \right] \end{equation*}
we have
\begin{equation*} \begin{array}{ccc} a_{1 2} \amp = \amp 0\\ a_{2 1} \amp = \amp 3\\ a_{3 1} \amp = \amp 6\\ a_{3 2} \amp = \amp 7\\ \end{array} \end{equation*}
Find the given elements for the matrix.
\begin{equation*} \boldsymbol{A} = \begin{array}{lrrrr} \lceil \amp {-5} \amp {-2} \amp {2} \amp \rceil\\ \vert \amp{2} \amp {7} \amp {7} \amp \vert \\ \vert \amp {1} \amp {-4} \amp {-8} \amp \vert\\ \lfloor \amp {8} \amp {-5} \amp {-9} \amp \rfloor\\ \end{array} \end{equation*}
  1. \(a_{3 1} =\)
  2. \(a_{4 3} =\)
  3. \(a_{1 2} =\)
  4. \(a_{2 2} =\)
  5. \(a_{4 1} =\)
Answer 1.
\(1\)
Answer 2.
\(-9\)
Answer 3.
\(-2\)
Answer 4.
\(7\)
Answer 5.
\(8\)
Solution.
  1. \(a_{3 1}:\) row 3 - column 1 = 1
  2. \(a_{4 3}:\) row 4 - column 3 = -9
  3. \(a_{1 2}:\) row 1 - column 2 = -2
  4. \(a_{2 2}:\) row 2 - column 2 = 7
  5. \(a_{4 1}:\) row 4 - column 1 = 8

Subsubsection 1.1.3 Matrix equality

Two matrices are equal when they have the same numbers in corresponding entries.
\begin{equation*} \boldsymbol{A} = \boldsymbol{B} \hspace{1cm} \text{If and only if} \hspace{1cm} a_{ij}=b_{ij} \hspace{0.5cm} \text{for ALL } i \text{ and } j's. \end{equation*}

Subsubsection 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 1.1.5 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 1.1.6 Zero matrix

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