Order of a matrix

The governing definition is that the order of a matrix is written as the number of rows multiplied by the number of columns. In the given matrix, the first horizontal row contains 1, 2, and 3, while the second horizontal row contains 4, 5, and 6. Therefore, there are 2 rows. Each row contains 3 entries, so the matrix has 3 columns. Hence its order is 2 × 3, making option A correct. Option B reverses the conventional order by writing columns before rows. Options C and D are incorrect because they describe square matrices, whereas a square matrix must have equal numbers of rows and columns. This matrix is rectangular, not square.

The matrix has three horizontal rows: [1 2], [3 4], [5 6]. Thus its order is \(3\times2\) and the number of rows is 3. Option A (2) confuses columns with rows; options C and D are clearly incorrect counts. Exam tip: count the horizontal bracketed lines to find the number of rows (each horizontal entry list is one row).

For an \(m\times n\) matrix there are \(n\) columns. In this matrix each row has 4 entries and there are 2 rows, so its size is \(2\times4\) and it has 4 columns. Option A (2) is the number of rows, option D (8) is the total number of entries; option B (3) is unrelated. Exam tip: count entries in a single row or read the matrix size \((m\times n)\) to find the number of columns quickly.

The number of elements equals the number of rows times the number of columns. For a matrix with \(m\) rows and \(n\) columns the total elements are \(m\times n\), i.e. \(mn\). Option A (\(m+n\)) incorrectly adds rows and columns instead of multiplying them; B (\(m-n\)) and D (\(m/n\)) are also incorrect because subtraction or division do not count grid entries. Exam tip: for a square matrix of order \(n\times n\) the total elements are \(n^2\).

Total elements = rows × columns. Let columns = n. Then \(15 = 3 \times n\), so \(n = 15/3 = 5\). Thus 5 is correct. Option C (12) likely comes from mistakenly doing 15−3; option D (18) would correspond to 3×6 and does not match 15; option A (3) gives 3×3 = 9, not 15. Exam tip: Use elements = rows × columns and divide the total elements by the number of rows to find columns.

Total elements of a matrix = rows × columns. Given total elements 24 and columns 6, we have \(24 = m \times 6\), so rows \(m = 24/6 = 4\). Option B (6) confuses columns with rows and is therefore incorrect. Exam tip: To find rows, divide total number of elements by number of columns (rows = elements ÷ columns).

Number of elements in a matrix = (number of rows) \(\times\) (number of columns). For a \(1\times1\) matrix the count is \(1\times1=1\), so there is one element. Option C (2) would correspond to a \(1\times2\) or \(2\times1\) matrix, option D (11) is a concatenation error (not a product), and option A (0) is incorrect here. Exam tip: multiply rows by columns to get the total elements every time.

The matrix \(\begin{bmatrix}2&3&4\end{bmatrix}\) has \(1\) row and \(3\) columns. Hence, its order is \(1\times 3\).

The matrix has four rows (2, 3, 4, 5) and one column, so its order is \(4\times1\). Option A (\(1\times4\)) is the closest distractor but incorrect because it denotes a 1-row, 4-column matrix; here there are 4 rows. Exam tip: Always count rows first, then columns (rows × columns).