Types of matrices

Matrix \(C\) has size \(1\times3\) (one row and three columns). A matrix with only one row is a row matrix, so the correct answer is row matrix. Closest distractor: a column matrix would have one column (e.g. \(3\times1\)), which is not the case here; diagonal and identity matrices must be square, unlike this \(1\times3\) matrix. Exam tip: always check the dimensions (rows × columns); if rows = 1 it's a row matrix.

All entries of this matrix are zero, so it is a zero matrix. A matrix whose every entry is 0 is called the zero matrix — therefore B is correct. Identity matrices have 1s on the main diagonal (not true here). A scalar matrix is of the form λI; the zero matrix is the special case with λ=0, but conventionally it is identified as the zero matrix in classification questions. A column matrix has a single column, whereas the given matrix is 2×2. Exam tip: Check the entries first — if every entry is 0, the matrix is the zero matrix immediately.

Definition: A rectangular matrix has a different number of rows and columns. In general an order \(m\times n\) matrix is rectangular if \(m\ne n\). Option C is of order \(2\times5\) and since \(2\ne5\) it is rectangular. Options A, B and D are of orders \(2\times2\), \(3\times3\) and \(1\times1\) respectively; in each of these the number of rows equals the number of columns, so they are square matrices and not rectangular. Exam tip: To decide quickly, compare the two numbers in the order \(m\times n\); equal ⇒ square, unequal ⇒ rectangular.

Definition: A diagonal matrix has all off-diagonal entries equal to zero. In the given matrix the off-diagonal entries are 0 and the diagonal entries are 2 and 3, so it is a diagonal matrix. Closest distractor: an identity matrix must have 1s on the diagonal, which this matrix does not. Exam tip: quickly check whether all non-diagonal elements are zero to recognize a diagonal matrix.

All entries outside the main diagonal are zero while the diagonal entries are 2, 3 and 4 (nonzero). These properties define a diagonal matrix. The zero-matrix option is incorrect because a zero matrix requires every entry to be 0, which is not true here. It is also not a row or column matrix since those have dimensions 1×n or n×1 whereas this is 3×3. Exam tip: Check whether all off-diagonal entries are 0 to identify a diagonal matrix quickly.

An identity matrix has 1s on the main diagonal and 0s elsewhere. The given matrix is 3×3 with ones on its main diagonal and zeros in all other positions, so it is the identity matrix of order 3. The closest distractor, the zero matrix, is incorrect because a zero matrix has every entry equal to 0, whereas here diagonal entries are 1. Row and column matrices are of sizes 1×n or n×1 respectively, so they do not match a 3×3 identity matrix. Exam tip: quickly check the main diagonal — if all are 1 and others 0, it's an identity matrix.

A scalar matrix is of the form kI, where I is the identity matrix and k is a scalar. Thus every main diagonal entry equals k and every off-diagonal entry is zero. The closest distractor, row matrix, only means nonzero entries may lie in a single row and does not require equal diagonal entries or zero off-diagonals. Exam tip: Check if the matrix can be written as kI — if yes, it's a scalar matrix.

An identity matrix (commonly denoted \(I_n\)) is defined by having 1s on the main diagonal and 0s elsewhere, so the given description matches the identity matrix exactly. The closest distractor is the diagonal matrix — it has zeros off the main diagonal but its diagonal entries need not be 1; the identity matrix is just the special diagonal matrix with all diagonal entries equal to 1. The zero matrix has all entries 0 (including the diagonal), and a rectangular matrix refers to shape and may not be square, whereas an identity matrix must be square. Exam tip: verify the matrix is square and that diagonal entries are specifically 1 before selecting the answer.

Definition: a zero matrix has every entry equal to 0. In option A all four entries are 0, so it is the 2×2 zero matrix. The closest distractor B is the identity matrix with 1s on the diagonal, so it is not a zero matrix. Exam tip: verify every entry — a single nonzero entry immediately rules out the zero matrix.

An identity matrix of order 2 must have the form I₂ = [[1, 0], [0, 1]]. The defining rule places 1 on every position of the main diagonal and 0 in every other position. Option B has exactly this arrangement, so it is the correct answer. Option A is the 2 × 2 zero matrix because all its entries are zero. Option C has ones away from the main diagonal as well, so it is not an identity matrix. Option D has zeros on the main diagonal and ones off the diagonal; it is a permutation or exchange matrix, not I₂. Checking diagonal and off-diagonal entries makes the identification unambiguous.

The governing classification is the definition of a scalar matrix. A scalar matrix is a square diagonal matrix in which every main-diagonal entry is the same scalar and all off-diagonal entries are zero. Here A is 2 × 2, its diagonal entries are both 3, and its off-diagonal entries are 0. Therefore A = 3I₂ and option A is correct. It is not a row matrix because it has two rows, nor a column matrix because it has two columns. The phrase rectangular matrix is not the intended specific classification: the displayed matrix is square and, more precisely, scalar. The equal diagonal entries are the decisive feature.