State the size of \(A=\begin{pmatrix}1&4&0\\-2&5&7\end{pmatrix}\).

For \(B=\begin{pmatrix}6&-1\\0&3\\8&2\end{pmatrix}\), identify \(b_{31}\).

Is \(\begin{pmatrix}2\\-5\\1\end{pmatrix}\) a row vector or a column vector? Give its size.

Write the set notation for all real \(4\times2\) matrices.

For \(C=\begin{pmatrix}3&0&-4\\1&9&2\\7&6&5\end{pmatrix}\), find \(c_{12}\) and \(c_{23}\).

A matrix has entries \(a_{ij}=i+j\) for \(1\le i\le2\), \(1\le j\le3\). Write the matrix.

Decide whether \(\begin{pmatrix}1&0&2\\3&4&5\\6&7&8\end{pmatrix}\) is square, and state the value of \(n\) if its size is \(n\times n\).

A learner says a \(2\times5\) matrix has \(5\) rows and \(2\) columns. Correct the statement.

Let \(A\in\operatorname{Mat}_{2\times4}(\mathbb R)\). How many entries does \(A\) have, and what are the possible row and column indices?

Construct a \(2\times3\) matrix \(A\) with \(a_{ij}=2i-j\).

Explain why a column vector with \(5\) entries belongs to \(\operatorname{Mat}_{5\times1}(\mathbb R)\), not \(\operatorname{Mat}_{1\times5}(\mathbb R)\).

Given \(D=\begin{pmatrix}d_{11}&d_{12}\\d_{21}&d_{22}\\d_{31}&d_{32}\end{pmatrix}\), explain which symbol names the entry in the bottom-right position.

A data table stores displacement components \((x,y,z)\) for \(4\) particles, with one particle per row and one component per column. What matrix size represents the table, and what does entry \(a_{23}\) mean?

A matrix \(A\) has \(12\) entries and \(3\) rows. If it is not a row vector, what is its size?

Find all possible sizes of a real matrix with \(6\) entries that is not square. Use positive row and column counts.

For an \(m\times n\) matrix \(A\), the entry \(a_{47}\) is defined but \(a_{74}\) is not. Give one possible pair \((m,n)\) and explain the restrictions.

A matrix \(A\in\operatorname{Mat}_{m\times n}(\mathbb R)\) is both a row vector and a column vector. Determine \(m\) and \(n\).

Diagnose the error: for \(A=\begin{pmatrix}5&6&7\\8&9&10\end{pmatrix}\), a learner says \(a_{23}=6\).

Prove that an \(m\times n\) matrix is square exactly when \(m=n\).

A physics list records \(n\) time samples of three quantities \(s\), \(v\), and \(a\). One convention stores samples as rows; another stores quantities as rows. Compare the two matrix sizes and explain why the same data can have different notation.