Read, transpose and classify a matrix

A matrix is an ordered rectangular array

An m×n matrix has m rows and n columns. The entry aᵢⱼ is in row i, column j, with indices beginning at one. Equality requires the same dimensions and equality at every matching position. Contextual matrices also need consistent row/column labels and units

Given matrix

\(\displaystyle A=\begin{bmatrix} 2 & -1 & 4 \\ 0 & 3 & 5 \end{bmatrix}\)

For the displayed A, how many columns are there?

Given matrix

\(\displaystyle A=\begin{bmatrix} 2 & -1 & 4 \\ 0 & 3 & 5 \end{bmatrix}\)

For the matrix A, find a₂₃

Transpose interchanges the index roles

The transpose Aᵀ has n rows and m columns and (Aᵀ)ᵢⱼ=Aⱼᵢ. Each original row becomes a column; applying transpose twice recovers A.

Given matrix

\(\displaystyle A=\begin{bmatrix} 2 & -1 & 4 \\ 0 & 3 & 5 \end{bmatrix}\)

For the matrix A, find the entry in row 3, column 2 of Aᵀ

Some classifications overlap

Row matrices have one row; column matrices have one column. Square matrices have equal dimensions. A square matrix is symmetric when Aᵀ=A. A diagonal matrix has zeros off the main diagonal; the identity has ones on that diagonal. Upper/lower triangular matrices have zeros below/above it. A zero matrix can be rectangular. An identity matrix is also diagonal, triangular and symmetric

Which statement always holds?