Special Types of Matrices

Diagonal matrix

A square matrix is diagonal if every entry off the main diagonal is zero; the diagonal entries can be anything, including zero, as in $\begin{bmatrix} 4 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 7 \end{bmatrix}$.

Scalar matrix

A scalar matrix is a diagonal matrix with every diagonal entry equal to the same number $k$, such as $5I = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$.

Identity matrix

The identity matrix $I$ is the scalar matrix with $k = 1$. It behaves like 1 in multiplication: $AI = IA = A$. So every identity matrix is scalar and every scalar matrix is diagonal, but not the other way round.

Zero and triangular matrices

The zero matrix $O$ has every entry 0. An upper triangular matrix has zeros below the diagonal, and a lower triangular one has zeros above it. A triangular matrix's determinant is the product of its diagonal: $\det\begin{bmatrix} 2 & 1 & 4 \\ 0 & -3 & 5 \\ 0 & 0 & 1 \end{bmatrix} = 2 \times (-3) \times 1 = -6$.

Worked example: how special matrices multiply

Diagonal matrices multiply entry by entry along the diagonal. A scalar matrix $kI$ commutes with every square matrix of the same order, $kI \cdot M = M \cdot kI = kM$, and $\det\begin{bmatrix} 4 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 7 \end{bmatrix} = -28$.

Worked example: using a stated property

If $\begin{bmatrix} x+2 & y-3 \\ 0 & 4 \end{bmatrix}$ is a scalar matrix, the off-diagonal entry must be 0, so $y = 3$, and the diagonal entries must match, so $x + 2 = 4$ and $x = 2$.

Worked example: matching determinants

If $\begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$, then $3 - x^2 = 3 - 8 = -5$, so $x^2 = 8$ and $x = \pm 2\sqrt2$.

Common mistakes

• Calling any matrix with zeros on its diagonal a diagonal matrix. It's the off-diagonal entries that must be zero.
• Assuming a diagonal matrix must be scalar. Its diagonal entries can differ.
• Keeping only one root. $x^2 = 8$ gives two values of $x$.
• Thinking only the identity commutes with everything. Every scalar matrix $kI$ does.

Where this comes up in exams

The 2025-26 CBSE Class XII syllabus lists types of matrices, including the zero and identity matrix. JEE Main 2026 lists types of matrices, and JEE Advanced 2026 lists diagonal, symmetric and skew-symmetric matrices and their properties.

Checked against CBSE Mathematics curriculum, Classes XI-XII (2025-26), JEE (Main) 2026 syllabus (NTA) and JEE (Advanced) 2026 syllabus. Syllabi can change from year to year, so confirm with the latest official notification for your exam.

• Fit it into your week with the free timetable generator.
• Mark it off as you revise in the syllabus tracker.
• See how long you have left: JEE Main 2027 (Session 1) countdown and JEE Advanced 2027 countdown.