24 Matrices

A matrix is a rectangular arrangement of numbers in rows and columns
\[ \begin {pmatrix} a_{11} & a_{12} & \cdots \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots \cdots & a_{2n}\\ \vdots & \vdots & & \vdots \\ a_{m1} & a_{m2} & \cdots \cdots & a_{mn}\\ \end {pmatrix} \]

A matrix is identified by the number of rows and columns as:

A \(2\times 2\) has 2 rows and 2 columns.

A \(3\times 2\) matrix has 3 rows and 2 columns

\[ \begin {pmatrix} a&b\\ c&d\\ e&f\\ \end {pmatrix} \quad \text {This is a} \, 3\times 2 \, \text {matrix}\]

• The components in the matrix are called elements.
• An \(n\times n\) matrix is called a square matrix.
• 0 (zero) matrix is where all the elements in a matrix are zeros.

\[ \begin {pmatrix} 0 & 0 & \cdots \cdots & 0\\ 0 & 0 & \cdots \cdots & 0\\ \vdots & \vdots && \vdots \\ 0 & 0 & \cdots \cdots & 0\\ \end {pmatrix} \]

\(\bullet \) \(I =\) Identity matrix is a matrix where the leading diagonal elements are 1’s and zeros elsewhere.

E.g

\( \begin {pmatrix} 1&0\\ 0&1\\ \end {pmatrix} \) Identity matrix for a \(2\times 2\) matrix

\( \begin {pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end {pmatrix} \) Identity matrix for a \(3\times 3\) matrix.

\( \begin {pmatrix} 1 & 0 & \cdots & 0\\ 0 & 1 & \cdots & 0\\ \vdots & \vdots && \vdots \\ 0 & 0 & \cdots & 1\\ \end {pmatrix} \) Identity matrix for a \(n\times n\) matrix.

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