Q.Define diagonal and scalar matrices.

A matrix is a rectangular array of numbers (or real-valued functions) arranged in rows and columns, enclosed in square brackets. If a matrix $A$ has $m$ rows and $n$ columns, we say $A$ is of order $m \times n$, and write $A = [a_{ij}]_{m\times n}$, $1 \le i \le m,\ 1 \le j \le n$, where $a_{ij}$ is the entry common to the $i$th row and $j$th column.

The book classifies matrices by shape and pattern of entries:

• Row matrix: only one row, order $1 \times n$.
• Column matrix: only one column, order $m \times 1$.
• Zero (null/void) matrix $O$: every entry is $0$.
• Square matrix: number of rows = number of columns, order $n \times n$. The entries $a_{11}, a_{22}, \ldots, a_{nn}$ form the principal (main/leading) diagonal.
• Diagonal matrix: a square matrix in which every off-diagonal entry is $0$ (i.e. $a_{ij}=0$ whenever $i\ne j$); the diagonal entries themselves may be anything, including $0$.
• Scalar matrix: a diagonal matrix whose diagonal entries are all equal to the same constant $c$.
• Unit (identity) matrix $I_n$: a diagonal matrix whose diagonal entries are all $1$. Every unit matrix is a scalar matrix (with $c=1$), and every square zero matrix is a (trivial) scalar/diagonal matrix.
• Upper triangular matrix: a square matrix with every entry below the main diagonal equal to $0$, i.e. $a_{ij}=0$ for all $i>j$. …