Introduction to Matrices

(Notation & Terminology)

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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:

• \(m\) is the number of rows the matrix has
• \(n\) is the number of columns the matrix has

If a matrix \(A\) is of order \(m\times n\) we'll often indicate this using notation: \[A_{m,n}\] For example, consider the matrices \(A\) and \(B\) shown here: \[A = \begin{pmatrix} 2 & 1 & -5 \\ 0 & 7 & 8 \end{pmatrix} \quad B = \begin{pmatrix} -1 & 0 & 7 \\ 6 & 2 & -3 \\ -5 & 1 & 9 \end{pmatrix}\] We say that:

• \(A\) is a \(2\times 3\) "two by three" matrix, which we can write \(A_{2,3}\)
• \(B\) is a \(3\times 3\) "three by three" matrix, which we can write \(B_{3,3}\)

Square Matrices

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:

• \(i\) refers to the row the entry is in
• \(j\) refers to the column the entry is in

For instance, given the \(3\times 3\) matrix \(A\) defined as: \[A = \begin{pmatrix}3 & 0 & -1 \\ 2 & 7 & 5 \\ -4 & 6 & 1 \end{pmatrix}\] we can refer to the entry \(7\), which is on the second row and in the second column, as well as the entry \(6\) on the thord row and in the second column by writing: \[a_{22} = 7, \quad a_{32} = 6\]

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Generic Matrix Notation (for Square Matrices)

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.

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\(2\times 2\) 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.

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\(3\times 3\) Matrices

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: \[A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}\]

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\(4\times 4\) Matrices and more

When working \[A = \begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{pmatrix}\]

Zero Matrix

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: \[0_{2,2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}, \quad 0_{1,3} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}\] \[0_{3,3} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\]

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Row Matrix

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.

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Column Matrix

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.

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Identity Matrix

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 \(1\)).