Question bank — Matrix definition — rows, columns, order, elements

True — equality first demands identical order, and $2\times 3\ne 3\times 2$ are different shapes, so the comparison fails before we even look at the numbers.

False — the $\times$ in "order" is a label, not multiplication; $4\times 5$ is 4 rows tall, $5\times 4$ is 5 rows tall, so they are genuinely different grids.

True — a square is just the special rectangle where width equals height; "rectangular array" is the umbrella that square matrices sit under.

False — total elements $=m\times n=12$ has several factorisations ($1\times 12,2\times 6,3\times 4,4\times 3,6\times 2,12\times 1$), so the count alone does not fix the shape.

False — row index 3 exceeds the 2 available rows, so $a_{31}$ points to nothing; an index is only valid when $1\le i\le m$ and $1\le j\le n$.

True as an object, but it is treated as a matrix of order $1\times 1$; the brackets matter because it can now take part in matrix operations, unlike a loose scalar.

True — unless $i=j$ (a diagonal element), reading row-$i$-column-$j$ versus row-$j$-column-$i$ lands on two different grid cells; this swap is exactly what the transpose does to the whole matrix.

False — a row matrix is $1\times n$ (one horizontal strip) and a column matrix is $m\times 1$ (one vertical strip); they coincide only in the single case $1\times 1$.

Two errors: the convention is row first, column second, so $c_{23}$ is row 2, column 3 — the reader reversed the roles of both indices.

They counted the four numbers as "rows". It is one horizontal row of three columns, so the order is $1\times 3$; counting entries is not counting rows.

Standard mathematical notation is 1-based, so the top-left entry is $a_{11}$; the zero-start habit comes from programming arrays, not from matrix algebra.

"Square" is about shape ($m=n$), not element count; this matrix is $3\times 2$ with $3\ne 2$, so it is rectangular regardless of how many numbers it holds.

The value $3$ is correct here, but only because $i+j$ is symmetric; the reasoning is unsafe — they used $i=2,j=1$ instead of $i=1,j=2$, which would give a wrong answer for any non-symmetric formula like $i+2j$.

Equal order is necessary but not sufficient; the matrices must also match element-by-element, $a_{ij}=b_{ij}$ for every $i,j$, before we can call them equal.

A grid is two-dimensional, so one coordinate cannot pin a cell; we need (which row, which column) — two numbers — to locate any entry unambiguously.

It mirrors how we write the order $m\times n$ (rows before columns) and how the general form is laid out, keeping one consistent left-to-right, top-to-bottom reading rule everywhere.

Order is the "shape passport": addition needs identical orders and multiplication needs the inner dimensions to match, so shape decides in advance which operations are even legal.

Because transformations are applied as matrix-times-column, $A\mathbf{x}$; writing the point as a column $m\times 1$ makes the multiplication line up dimensionally on the right.

The definition asks only for a rectangular arrangement of numbers or expressions; the grid structure is what makes it a matrix, not whether each cell is already simplified to a bare number.

A single matrix lets us apply one set of systematic rules (determinant, row operations, solving systems) to all equations at once, instead of juggling each equation separately.

In standard school-level treatment, no — a matrix must have at least one row and one column ($m\ge 1,n\ge 1$); an "empty" grid holds no elements and is not used here.

A single number in brackets, order $1\times 1$; it is simultaneously the smallest square, row, and column matrix.

Yes — the zero matrix is perfectly valid; "matrix" describes the arrangement, and there is no rule forbidding the entries from all being zero (see types of matrices).

Exactly $n$ — these are the diagonal elements $a_{11},a_{22},\dots ,a_{nn}$ running from top-left to bottom-right.

Always $1$, since there is only one row; only the column index $j$ varies from $1$ to $n$, which is why row vectors are often written with a single subscript in practice.

Only when $i=j$ (a diagonal position); off the diagonal they are different cells whose values may or may not coincide.

Yes — widening a row increases the column count $n$, so a $2\times 3$ becomes $2\times 4$; you cannot change element counts without changing the order.