Q.The matrix $P = \begin{bmatrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{bmatrix}$ is a
(A) square matrix
(B) diagonal matrix
(C) unit matrix
(D) none

The key idea is to check the definitions: a square matrix has equal rows and columns, a diagonal matrix has non-zero entries only on the main diagonal, and a unit matrix (identity matrix) has 1's on the diagonal and 0's elsewhere.

• $P$ is $3 \times 3$, so it is a square matrix — option (A) is true. …

The question asks us to classify the given matrix $P$ among the listed types. Let’s first recall what each term means, then check $P$ against each definition.

Square matrix: A matrix with the same number of rows and columns. $P$ is $3 \times 3$, so it is square. This is the most basic property — every diagonal or unit matrix is automatically square, but not every square matrix is diagonal or unit.

Diagonal matrix: A square matrix where all entries outside the main diagonal are zero. The main diagonal runs from top‑left to bottom‑right: positions $(1,1)$, $(2,2)$, $(3,3)$. In $P$, the off‑diagonal entries are $4$ at $(1,3)$ and $(3,1)$ — these are not zero. So $P$ is not diagonal.

Unit matrix (also called identity matrix): A diagonal matrix where every diagonal entry is $1$. $P$ fails on two counts: it is not diagonal, and its diagonal entries are $0,4,0$, not all $1$. So it is not a unit matrix.

Since $P$ is square but not diagonal and not unit, the only correct classification among the given options is “square matrix”. …

Method: Classifying a matrix by checking definitions in order

For "the matrix is a (square / diagonal / identity / none)" questions, test the candidate definitions in increasing strictness and pick the strongest one that genuinely applies.

Steps

Step 1: Check "square".

A matrix is square if it has the same number of rows and columns. A $3\times3$ matrix is automatically square.

Step 2: Check "diagonal".

A diagonal matrix has ALL off-diagonal entries equal to $0$. Inspect every non-diagonal position — a single nonzero off-diagonal entry disqualifies it.

Step 3: Check "identity / unit". …

Common Mistakes

Mistake 1: Calling the matrix diagonal because its nonzero entries form a neat pattern.

Why it's wrong: the nonzero $4$s sit at positions $(1,3)$ and $(3,1)$ — off the main diagonal — so it is NOT diagonal. Correct approach: a diagonal matrix must have zeros in every off-diagonal position, including the anti-diagonal.

Mistake 2: Choosing "none" after ruling out diagonal and identity. …