A matrix is a rectangular array of numbers, written between a large pair of parentheses or square brackets, and usually named by a capital letter. For example: \[A = \begin{pmatrix} 2 & -1 & 0 \\ 3 & 5 & 6\end{pmatrix}, \quad B = \begin{bmatrix} 2 & 5 \\ -1 & 0 \end{bmatrix}\] \[C = \begin{pmatrix} 2 & 3 & -1 \\ 0 & 4 & 5 \\ 1 & 7 & -8 \end{pmatrix}\] are all matrices.
Whether you write a matrix between square brackets or large parenthese makes no difference whatsoever, both notations are widely accepted.
Given a matrix, its order, or size, is written: \[m\times n\] where:
If a matrix has same number of rows as columns then we say it is a square matrix. We'll come across square matrices a lot in this course. Here are a few examples: \[A_2 = \begin{pmatrix} 3 & 4 \\ -1 & 5 \end{pmatrix}, \quad B_3 = \begin{pmatrix} 1 & -5 & 3 \\ -4 & 2 & 6 \\ 0 & 8 & 9\end{pmatrix}\] \[C_4 = \begin{pmatrix}1 & 0 & -1 & 1\\ 3 & 5 & 7 & -9\\ -2 & 4 & -6 & 8 \\ 1 & -1 & 0 & 1\end{pmatrix}\] Notice that when dealing with square
To refer to specific entries inside a matrix we'll often use index notation. With index notation we can refer to any specific entry using the notation: \[a_{ij}\] where:
When working with matrices it's important to be familiar/comfortable with matrix notation. In particular when deriving/learning formula we'll work with generic matrices.
When working with \(2\times 2\) matrices we'll often use the following generic matrix \(A\): \[A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\] where \(a\), \(b\), \(c\) and \(d\) can be any real numbers.
When working with \(3\times 3\) matrices we'll need to use index notation, \(a_{ij}\) for each of the entries in the matrix. The typical matrix we'll use is: \[A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}\]
When working \[A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix}\]
A zero matrix, also called null matrix, has all entries equal to \(0\), and is often referred to with the notation \(0_{m,n}\). Here are some zero matrices: \[0_{2,2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}, \quad 0_{1,3} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}\] \[0_{3,3} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\]
A row matrix only has one row: \[A_{1,n} = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \end{pmatrix}\] Here are some examples of row matrices: \[A = \begin{pmatrix} -2 & 1 & 5 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & -6 \end{pmatrix}\] Note: row matrices can be thought of as row vectors, in fact we'll often treat them that way as we learn more about matrices.
A column matrix only has one column: \[A_{m,1} = \begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix}\] Here are some examples of column matrices: \[A = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}, \quad B = \begin{pmatrix} 6 \\ 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 \\ 3 \\ -1 \\ 2 \end{pmatrix}\] Note: column matrices can be thought of as column vectors, in fact we'll often treat them that way as we learn more about matrices.
An identity matrix is a square matrix whose diagonal entries are all equal to \(1\) and all other entries are equal to \(0\). Here are some identity matrices: \[I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\] \[I_4 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\] as we'll be seeing, when we multiply a matrix by an identity matrix it doesn't change its value (just like when we multiply a number by \(1\)).
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